Run-to-run control method for proportional-integral-derivative (PID) controller tuning for rapid thermal processing (RTP)

ABSTRACT

A method is provided, the method comprising measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step, and modeling the at least one characteristic parameter measured using a first-principles radiation model. The method also comprises applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to semiconductor fabrication technology, and, more particularly, to a method for semiconductor fabrication supervision and optimization.

2. Description of the Related Art

There is a constant drive within the semiconductor industry to increase the quality, reliability and throughput of integrated circuit devices, e.g., microprocessors, memory devices, and the like. This drive is fueled by consumer demands for higher quality computers and electronic devices that operate more reliably. These demands have resulted in a continual improvement in the manufacture of semiconductor devices, e.g., transistors, as well as in the manufacture of integrated circuit devices incorporating such transistors. Additionally, reducing defects in the manufacture of the components of a typical transistor also lowers the overall cost per transistor as well as the cost of integrated circuit devices incorporating such transistors.

The technologies underlying semiconductor processing tools have attracted increased attention over the last several years, resulting in substantial refinements. However, despite the advances made in this area, many of the processing tools that are currently commercially available suffer certain deficiencies. In particular, such tools often lack advanced process data monitoring capabilities, such as the ability to provide historical parametric data in a user-friendly format, as well as event logging, real-time graphical display of both current processing parameters and the processing parameters of the entire run, and remote, i.e., local site and worldwide, monitoring. These deficiencies can engender nonoptimal control of critical processing parameters, such as throughput accuracy, stability and repeatability, processing temperatures, mechanical tool parameters, and the like. This variability manifests itself as within-run disparities, run-to-run disparities and tool-to-tool disparities that can propagate into deviations in product quality and performance, whereas an ideal monitoring and diagnostics system for such tools would provide a means of monitoring this variability, as well as providing means for optimizing control of critical parameters.

Among the parameters it would be useful to monitor and control are process parameters related to rapid thermal processing (RTP). Examples of such process parameters include the temperatures and lamp power levels that silicon wafers and/or workpieces are exposed to during the rapid thermal processing (RTP) used to activate dopant implants, for example. The rapid thermal processing (RTP) performance typically degrades with consecutive process runs, in part due to drift in the respective settings of the rapid thermal processing (RTP) tool and/or the rapid thermal processing (RTP) sensors. This may cause differences in wafer processing between successive runs or batches or lots of wafers, leading to decreased satisfactory wafer throughput, decreased reliability, decreased precision and decreased accuracy in the semiconductor manufacturing process.

However, traditional statistical process control (SPC) techniques are often inadequate to control precisely process parameters related to rapid thermal processing (RTP) in semiconductor and microelectronic device manufacturing so as to optimize device performance and yield. Typically, statistical process control (SPC) techniques set a target value, and a spread about the target value, for the process parameters related to rapid thermal processing (RTP). The statistical process control (SPC) techniques then attempt to minimize the deviation from the target value without automatically adjusting and adapting the respective target values to optimize the semiconductor device performance, and/or to optimize the semiconductor device yield and throughput. Furthermore, blindly minimizing non-adaptive processing spreads about target values may not increase processing yield and throughput.

Traditional control techniques are frequently ineffective in reducing off-target processing and in improving sort yields. For example, the wafer electrical test (WET) measurements are typically not performed on processed wafers until quite a long time after the wafers have been processed, sometimes not until weeks later. When one or more of the processing steps are producing resulting wafers that wafer electrical test (WET) measurements indicate are unacceptable, causing the resulting wafers to be scrapped, this misprocessing goes undetected and uncorrected for quite a while, often for weeks, leading to many scrapped wafers, much wasted material and decreased overall throughput.

The present invention is directed to overcoming, or at least reducing the effects of, one or more of the problems set forth above.

SUMMARY OF THE INVENTION

In one aspect of the present invention, a method is provided, the method comprising measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step, and modeling the at least one characteristic parameter measured using a first-principles radiation model. The method also comprises applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.

In another aspect of the present invention, a computer-readable, program storage device is provided, encoded with instructions that, when executed by a computer, perform a method, the method comprising measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step, and modeling the at least one characteristic parameter measured using a first-principles radiation model. The method also comprises applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.

In yet another aspect of the present invention, a computer programmed to perform a method is provided, the method comprising measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step, and modeling the at least one characteristic parameter measured using a first-principles radiation model. The method also comprises applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.

In another aspect of the present invention, a system is provided, the system comprising a tool for measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step, and a computer for modeling the at least one characteristic parameter measured using a first-principles radiation model. The system also comprises a controller for applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.

In yet another aspect of the present invention, a device is provided, the device comprising means for measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step, and means for modeling the at least one characteristic parameter measured using a first-principles radiation model. The device also comprises means for applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention may be understood by reference to the following description taken in conjunction with the accompanying drawings, in which the leftmost significant digit(s) in the reference numerals denote(s) the first figure in which the respective reference numerals appear, and in which:

FIGS. 1-15 schematically illustrate various embodiments of a method for manufacturing according to the present invention; and, more particularly:

FIGS. 1, and 3-10 schematically illustrate a flow chart for various illustrative embodiments of a method according to the present invention;

FIG. 2 schematically illustrates in cross-section an AST SHS 2800 rapid thermal processing (RTP) tool representative of those used in various illustrative embodiments of the present invention;

FIG. 11 schematically illustrates a method for fabricating a semiconductor device practiced in accordance with the present invention;

FIG. 12 schematically illustrates workpieces being processed using a rapid thermal processing (RTP) tool, using a plurality of control input signals, in accordance with the present invention;

FIGS. 13-14 schematically illustrate one particular embodiment of the process and tool in FIG. 12;

FIG. 15 schematically illustrates one particular embodiment of the method of FIG. 11 that may be practiced with the process and tool of FIGS. 13-14;

FIGS. 16 and 17 schematically illustrate first and second Principal Components for respective rapid thermal processing data sets;

FIGS. 18 and 19 schematically illustrate geometrically Principal Components Analysis for respective rapid thermal processing data sets; and

FIGS. 20-23 schematically illustrate geometrically polynomial least-squares fitting, in accordance with the present invention.

While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof have been shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the description herein of specific embodiments is not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

Illustrative embodiments of the invention are described below. In the interest of clarity, not all features of an actual implementation are described in this specification. It will of course be appreciated that in the development of any such actual embodiment, numerous implementation-specific decisions must be made to achieve the developers' specific goals, such as compliance with system-related and business-related constraints, which will vary from one implementation to another. Moreover, it will be appreciated that such a development effort might be complex and time-consuming, but would nevertheless be a routine undertaking for those of ordinary skill in the art having the benefit of this disclosure.

Illustrative embodiments of a method according to the present invention are shown in FIGS. 1-23. As shown in FIG. 1, a workpiece 100, such as a semiconducting substrate or wafer, having zero, one or more process layers and/or semiconductor devices, such as a metal-oxide-semiconductor (MOS) transistor disposed thereon, is delivered to a rapid thermal processing (RTP) step 105. In the rapid thermal processing (RTP) step 105, rapid thermal processing, such as a rapid thermal anneal, may be performed on the workpiece 100.

FIG. 2 schematically illustrates in cross-section a rapid thermal anneal (RTA) tool 200, e.g., an AST SHS 2800 rapid thermal anneal (RTA) tool, that may be used in the rapid thermal processing (RTP) step 105 in various illustrative embodiments according to the present invention. Various alternative illustrative embodiments of the present invention may use rapid thermal anneal (RTA) tools (such as the Centura® RTP) manufactured by Applied Materials (AMAT), which are quite different in physical form, usage, and measured parameters, but which may, nevertheless, be used in the rapid thermal processing (RTP) step 105.

As shown in FIG. 2, the illustrative rapid thermal anneal (RTA) tool 200 may heat a workpiece 100, such as a semiconducting silicon wafer with zero, one, or more process layers formed thereon, by using an array of halogen lamps 210 disposed above and below the workpiece 100. The workpiece 100 may be disposed on quartz pins and a wafer stand 215 within a quartz tube 220 heated by the array of halogen lamps 210. The wafer stand 215 may include other components, such as an AST Hot Liner™. The temperature of the quartz tube 220 may be measured by a thermocouple and/or a pyrometer 230 that measures the temperature of the AST Hot Liner™ component of the wafer stand 215 and/or a separate pyrometer (not shown). The quartz tube 220 may have a quartz window 225 disposed therein below the wafer stand 215. The temperature of the AST Hot Liner™ component of the wafer stand 215, and, indirectly, the workpiece 100 may be measured through the quartz window 225 by the pyrometer 230 disposed below the quartz window 225. Alternatively, the pyrometer 230 disposed below the quartz window 225 may directly measure the temperature of the workpiece 100. The lamp power of the halogen lamps 210 may also be monitored and controlled.

As shown in FIG. 3, the rapid thermal processing (RTP) step 105 may communicate with a monitoring step 110 and other processing steps 140 via bidirectional connections through a system communications bus 160. As shown in FIG. 3, the system communications bus 160 also provides communications between the rapid thermal processing (RTP) step 105, the monitoring step 110 and other processing steps 140, and an Advanced Process Control (APC) system 120, more fully described below.

As shown in FIG. 4, the workpiece 100 is sent from the rapid thermal processing (RTP) step 105 and delivered to the monitoring step 110. In the monitoring step 110, one or more rapid thermal processing (RTP) tool variables and/or one or more rapid thermal processing (RTP) parameters during one or more rapid thermal processing (RTP) runs may be monitored and/or measured. Examples of such tool variables and/or rapid thermal processing (RTP) parameters may comprise one or more pyrometer trace readings, one or more lamp power trace readings, one or more tube temperature trace readings, gas composition, gas flow rates, pressure, and the like. As shown in FIG. 4, the monitoring step 110 may communicate with the rapid thermal processing (RTP) step 105 via the system communications bus 160. As shown in FIG. 4, the system communications bus 160 also provides communications between the rapid thermal processing (RTP) step 105, the monitoring step 110, and the Advanced Process Control (APC) system 120, more fully described below.

As shown in FIG. 5, the workpiece 100 is sent from the monitoring step 110 and delivered to the other processing steps 140. In the other processing steps 140, other processing steps may be performed on the workpiece 100 to produce the finished workpiece 100. In alternative illustrative embodiments, the workpiece 100 sent from the monitoring step 110 may be the finished workpiece 100, in which case, there may not be other processing steps 140. As shown in FIG. 5, the other processing steps 140 may communicate with the monitoring step 110 via the system communications bus 160. As shown in FIG. 5, the system communications bus 160 also provides communications between the monitoring step 110, the other processing steps 140, and the Advanced Process Control (APC) system 120, more fully described below.

As shown in FIG. 6, the monitored sensor data 115 is sent from the monitoring step 110 and delivered to the Advanced Process Control (APC) system 120. As shown in FIG. 6, the Advanced Process Control (APC) system 120 may communicate with the monitoring step 110 via the system communications bus 160. Delivering the monitored sensor data 115 to the Advanced Process Control (APC) system 120 produces an output signal 125.

As shown in FIG. 7, the output signal 125 is sent from the Advanced Process Control (APC) system 120 and delivered to a first-principles radiation modeling with proportional-integral-derivative (PID) tuning step 130. In the first-principles radiation modeling with proportional-integral-derivative (PID) tuning step 130, the monitored sensor data 115 may be used in a first-principles radiation model, appropriate for the rapid thermal processing (RTP) performed on the workpiece 100 in the rapid thermal processing (RTP) step 105. The use of the monitored sensor data 115 in a first-principles radiation model produces one or more rapid thermal processing (RTP) recipe adjustments 145.

In various illustrative embodiments, a first-principles radiation model may be built. Such a first-principles radiation model may also be formed by monitoring one or more rapid thermal processing (RTP) tool variables and/or one or more rapid thermal processing (RTP) parameters during one or more rapid thermal processing (RTP) runs. As described above, examples of such rapid thermal processing (RTP) tool variables and/or rapid thermal processing (RTP) parameters may comprise one or more pyrometer trace readings, one or more lamp power trace readings, one or more tube temperature trace readings, gas composition, gas flow rates, pressure, and the like. In these various illustrative embodiments, building the first-principles radiation models may comprise fitting the collected processing data using at least one of polynomial curve fitting, least-squares fitting, polynomial least-squares fitting, non-polynomial least-squares fitting, weighted least-squares fitting, weighted polynomial least-squares fitting, weighted non-polynomial least-squares fitting, Partial Least Squares (PLS), and Principal Components Analysis (PCA), as described more fully below.

In various illustrative embodiments, the first-principles radiation model may incorporate at least one proportional-integral-derivative (PID) controller having at least one tuning parameter. In various of these illustrative embodiments, the first-principles radiation model, appropriate for rapid thermal processing (RTP), may incorporate at least one closed-loop proportional-integral-derivative (PID) controller having at least one tuning parameter. The proportional-integral-derivative (PID) controller tuning parameter(s) may be optimized based on an objective function that minimizes undesirable rapid thermal processing (RTP) conditions in the rapid thermal processing (RTP) performed on the workpiece 100 in the rapid thermal processing (RTP) step 105. For example, the difference(s) between a setpoint trace (temperature plotted against time) and a pyrometer trace (temperature plotted against time) may be minimized.

The proportional-integral-derivative (PID) controller may be designed to generate an output that causes some corrective effort to be applied to the rapid thermal processing (RTP) performed on the workpiece 100 in the rapid thermal processing (RTP) step 105 to drive one or more measurable rapid thermal processing (RTP) tool variable and/or one or more rapid thermal processing (RTP) parameter toward a respective desired value known as the setpoint. The proportional-integral-derivative (PID) controller may generate the output that causes the corrective effort by monitoring and/or measuring and/or observing the error between the setpoint and a measurement of the respective rapid thermal processing (RTP) tool variable(s) and/or rapid thermal processing (RTP) parameter(s).

The proportional-integral-derivative (PID) controller may look at the current value of the error e(t), the integral of the error e(t) over a recent time interval, and the current value of the derivative of the error e(t) with respect to time to determine how much of a correction to apply and for how long. Multiplying each of these terms by a respective tuning constant and adding them together generates the proportional-integral-derivative (PID) controller current output CO(t) given by the expression ${{{CO}(t)} = {{P\left( {e(t)} \right)} + {I\left( {\int{{e(t)}{t}}} \right)} + {D\left( {\frac{}{t}{e(t)}} \right)}}},$

where P is the proportional tuning constant, I is the integral tuning constant, D is the derivative tuning constant, and the error e(t) is the difference between the setpoint SP(t) and the process variable PV(t) at time t, e(t)=SP(t)−PV(t). If the current error e(t) is large and/or the error e(t) has been large for a long time and/or the current error e(t) is changing rapidly, the current controller output CO(t) may also be large. However, if the current error e(t) is small, the error e(t) has been small for a long time, and the current error e(t) is changing slowly, the current controller output CO(t) may also be small.

In various alternative illustrative embodiments, the proportional-integral-derivative (PID) controller current output CO(t) may be given by the alternative expression ${{{CO}(t)} = {P\left\lbrack {{e(t)} + {\frac{1}{T_{I}}\left( {\int{{e(t)}{t}}} \right)} - {T_{D}\left( {\frac{}{t}{{PV}(t)}} \right)}} \right\rbrack}},$

where P is an overall tuning constant, T_(I) is the integral time tuning constant, T_(D) is the derivative time tuning constant, and the error e(t) is the difference between the setpoint SP(t) and the process variable PV(t) at time t, e(t)=SP(t)−PV(t). In these alternative illustrative embodiments, there are fewer abrupt changes in the proportional-integral-derivative (PID) controller current output CO(t) when there is a change to the setpoint SP(t), due to the dependence on the time derivative of the process variable PV(t), rather than on the time derivative of the error e(t)=SP(t)−PV(t).

The proportional-integral-derivative (PID) controller current output CO(t) tuning constants P, I, and D, and/or P, T_(I), and T_(D), may be tuned appropriately. Using aggressively large values for the tuning constants P, I, and D, and/or P, T_(I), and T_(D), may amplify the error e(t) and overcompensate and overshoot the setpoint(s). Using conservatively small values for the tuning constants P, I, and D, and/or P, T_(I), and T_(D), may reduce the error e(t) too slowly and undercompensate and undershoot the setpoint(s). Appropriately tuned proportional-integral-derivative (PID) controller current output CO(t) tuning constants P, I, and D, and/or P, T_(I), and T_(D), may lie between these two extremes. The proportional-integral-derivative (PID) controller current output CO(t) tuning constants P, I, and D, and/or P, T_(I), and T_(D), may be tuned appropriately using trial-and-error tweaking, using a more rigorous analytical approach involving mathematical modeling, as described more fully below, and/or using techniques such as the Ziegler-Nichols “open loop” and “closed loop” tuning techniques.

The first-principles radiation modeling of the monitored sensor data 115 in the first-principles radiation modeling with proportional-integral-derivative (PID) tuning step 130, may be used to alert an engineer of the need to adjust the processing performed in any of a variety of processing steps, such as the rapid thermal processing (RTP) step 105 and/or the other processing steps 140. The engineer may also alter and/or adjust, for example, the setpoints for the rapid thermal processing (RTP) performed in the rapid thermal processing (RTP) step 105, and/or the rapid thermal processing (RTP) tool variable(s) and/or rapid thermal processing (RTP) parameter(s) monitored and/or measured in the monitoring step 110.

As shown in FIG. 8, a feedback control signal 135 may be sent from the first-principles radiation modeling with proportional-integral-derivative (PID) tuning step 130 to the rapid thermal processing (RTP) step 105 to adjust the rapid thermal processing (RTP) performed in the rapid thermal processing (RTP) step 105. In various alternative illustrative embodiments, the feedback control signal 135 may be sent from the first-principles radiation modeling with proportional-integral-derivative (PID) tuning step 130 to any of the other processing steps 140 to adjust the processing performed in any of the other processing steps 140, for example, via the system communications bus 160 that provides communications between the rapid thermal processing (RTP) step 105, the monitoring step 110, the other processing steps 140, and the Advanced Process Control (APC) system 120, more fully described below.

As shown in FIG. 9, in addition to, and/or instead of, the feedback control signal 135, the one or more rapid thermal processing (RTP) recipe adjustments 145, and/or an entire appropriate recipe based upon this analysis, may be sent from the first-principles radiation modeling with proportional-integral-derivative (PID) tuning step 130 to a rapid thermal processing (RTP) process change and control step 150. In the rapid thermal processing (RTP) process change and control step 150, the one or more rapid thermal processing (RTP) recipe adjustments 145 may be used in a high-level supervisory control loop. Thereafter, as shown in FIG. 10, a feedback control signal 155 may be sent from the rapid thermal processing (RTP) process change and control step 150 to the rapid thermal processing (RTP) step 105 to adjust the rapid thermal processing (RTP) performed in the rapid thermal processing (RTP) step 105. In various alternative illustrative embodiments, the feedback control signal 155 may be sent from the rapid thermal processing (RTP) process change and control step 150 to any of the other processing steps 140 to adjust the processing performed in any of the other processing steps 140, for example, via the system communications bus 160 that provides communications between the rapid thermal processing (RTP) step 105, the monitoring step 110, the other processing steps 140, and the Advanced Process Control (APC) system 120, more fully described below.

In various illustrative embodiments, the engineer may be provided with advanced process data monitoring capabilities, such as the ability to provide historical parametric data in a user-friendly format, as well as event logging, real-time graphical display of both current processing parameters and the processing parameters of the entire run, and remote, i.e., local site and worldwide, monitoring. These capabilities may engender more optimal control of critical processing parameters, such as throughput accuracy, stability and repeatability, processing temperatures, mechanical tool parameters, and the like. This more optimal control of critical processing parameters reduces this variability. This reduction in variability manifests itself as fewer within-run disparities, fewer run-to-run disparities and fewer tool-to-tool disparities. This reduction in the number of these disparities that can propagate means fewer deviations in product quality and performance. In such an illustrative embodiment of a method of manufacturing according to the present invention, a monitoring and diagnostics system may be provided that monitors this variability and optimizes control of critical parameters.

FIG. 11 illustrates one particular embodiment of a method 1100 practiced in accordance with the present invention. FIG. 12 illustrates one particular apparatus 1200 with which the method 1100 may be practiced. For the sake of clarity, and to further an understanding of the invention, the method 1100 shall be disclosed in the context of the apparatus 1200. However, the invention is not so limited and admits wide variation, as is discussed further below.

Referring now to both FIGS. 11 and 12, a batch or lot of workpieces or wafers 1205 is being processed through a rapid thermal processing (RTP) tool 1210. The rapid thermal processing (RTP) tool 1210 may be any rapid thermal processing (RTP) tool known to the art, provided it comprises the requisite control capabilities. The rapid thermal processing (RTP) tool 1210 comprises a rapid thermal processing (RTP) tool controller 1215 for this purpose. The nature and function of the rapid thermal processing (RTP) tool controller 1215 will be implementation specific.

For instance, the rapid thermal processing (RTP) tool controller 1215 may control rapid thermal processing (RTP) control input parameters such as rapid thermal processing (RTP) recipe control input parameters and/or setpoints. Four workpieces 1205 are shown in FIG. 12, but the lot of workpieces or wafers, i.e., the “wafer lot,” may be any practicable number of wafers from one to any finite number.

The method 1100 begins, as set forth in box 1120, by measuring one or more parameters characteristic of the rapid thermal processing (RTP) performed on the workpiece 1205 in the rapid thermal processing (RTP) tool 1210. The nature, identity, and measurement of characteristic parameters will be largely implementation specific and even tool specific. For instance, capabilities for monitoring process parameters vary, to some degree, from tool to tool. Greater sensing capabilities may permit wider latitude in the characteristic parameters that are identified and measured and the manner in which this is done. Conversely, lesser sensing capabilities may restrict this latitude. In turn, the rapid thermal processing (RTP) control input parameters such as the rapid thermal processing (RTP) recipe control input parameters and/or the setpoints for workpiece temperature and/or lamp power and/or anneal time and/or ramp rates and/or anneal recipes and/or the gas environment may directly affect the effective yield of usable semiconductor devices from the workpiece 1205.

Turning to FIG. 12, in this particular embodiment, the rapid thermal processing (RTP) process characteristic parameters are measured and/or monitored by tool sensors (not shown). The outputs of these tool sensors are transmitted to a computer system 1230 over a line 1220. The computer system 1230 analyzes these sensor outputs to identify the characteristic parameters.

Returning, to FIG. 11, once the characteristic parameter is identified and measured, the method 1100 proceeds by modeling the measured and identified characteristic parameter(s) using a first-principles radiation model, as set forth in box 1130. The computer system 1230 in FIG. 12 is, in this particular embodiment, programmed to model the characteristic parameter(s). The manner in which this modeling occurs will be implementation specific.

In the embodiment of FIG. 12, a database 1235 stores a plurality of models that might potentially be applied, depending upon which characteristic parameter is measured. This particular embodiment, therefore, requires some a priori knowledge of the characteristic parameters that might be measured. The computer system 1230 then extracts an appropriate model from the database 1235 of potential models to apply to the measured characteristic parameters. If the database 1235 does not include an appropriate model, then the characteristic parameter may be ignored, or the computer system 1230 may attempt to develop one, if so programmed. The database 1235 may be stored on any kind of computer-readable, program storage medium, such as an optical disk 1240, a floppy disk 1245, or a hard disk drive (not shown) of the computer system 1230. The database 1235 may also be stored on a separate computer system (not shown) that interfaces with the computer system 1230.

Modeling of the measured characteristic parameter may be implemented differently in alternative embodiments. For instance, the computer system 1230 may be programmed using some form of artificial intelligence to analyze the sensor outputs and controller inputs to develop a model on-the-fly in a real-time implementation. This approach might be a useful adjunct to the embodiment illustrated in FIG. 12, and discussed above, where characteristic parameters are measured and identified for which the database 1235 has no appropriate model.

The method 1100 of FIG. 11 then proceeds by applying the model to modify at least one rapid thermal processing (RTP) control input parameter, as set forth in box 1140. Depending on the implementation, applying the model may yield either a new value for a rapid thermal processing (RTP) control input parameter or a correction to an existing rapid thermal processing (RTP) control input parameter. In various illustrative embodiments, a multiplicity of control input recipes may be stored and an appropriate one of these may be selected based upon one or more of the determined parameters. The new rapid thermal processing (RTP) control input is then formulated from the value yielded by the model and is transmitted to the rapid thermal processing (RTP) tool controller 1215 over the line 1220. The rapid thermal processing (RTP) tool controller 1215 then controls subsequent rapid thermal processing (RTP) process operations in accordance with the new rapid thermal processing (RTP) control inputs.

Some alternative embodiments may employ a form of feedback to improve the modeling of characteristic parameters. The implementation of this feedback is dependent on several disparate facts, including the tool's sensing capabilities and economics. One technique for doing this would be to monitor at least one effect of the model's implementation and update the model based on the effect(s) monitored. The update may also depend on the model. For instance, a linear model may require a different update than would a non-linear model, all other factors being the same.

As is evident from the discussion above, some features of the present invention may be implemented in software. For instance, the acts set forth in the boxes 1120-1140 in FIG. 11 are, in the illustrated embodiment, software-implemented, in whole or in part. Thus, some features of the present invention are implemented as instructions encoded on a computer-readable, program storage medium. The program storage medium may be of any type suitable to the particular implementation. However, the program storage medium will typically be magnetic, such as the floppy disk 1245 or the computer 1230 hard disk drive (not shown), or optical, such as the optical disk 1240. When these instructions are executed by a computer, they perform the disclosed functions. The computer may be a desktop computer, such as the computer 1230. However, the computer might alternatively be a processor embedded in the rapid thermal processing (RTP) tool 1210. The computer might also be a laptop, a workstation, or a mainframe in various other embodiments. The scope of the invention is not limited by the type or nature of the program storage medium or computer with which embodiments of the invention might be implemented.

Thus, some portions of the detailed descriptions herein are, or may be, presented in terms of algorithms, functions, techniques, and/or processes. These terms enable those skilled in the art most effectively to convey the substance of their work to others skilled in the art. These terms are here, and are generally, conceived to be a self-consistent sequence of steps leading to a desired result. The steps are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electromagnetic signals capable of being stored, transferred, combined, compared, and otherwise manipulated.

It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, and the like. All of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities and actions. Unless specifically stated otherwise, or as may be apparent from the discussion, terms such as “processing,” “computing,” “calculating,” “determining,” “displaying,” and the like, used herein refer to the action(s) and processes of a computer system, or similar electronic and/or mechanical computing device, that manipulates and transforms data, represented as physical (electromagnetic) quantities within the computer system's registers and/or memories, into other data similarly represented as physical quantities within the computer system's memories and/or registers and/or other such information storage, transmission and/or display devices.

Construction of an Illustrative Apparatus. An exemplary embodiment 1300 of the apparatus 1200 in FIG. 12 is illustrated in FIGS. 13-14, in which the apparatus 1300 comprises a portion of an Advanced Process Control (“APC”) system. FIGS. 13-14 are conceptualized, structural and functional block diagrams, respectively, of the apparatus 1300. A set of processing steps is performed on a lot of workpieces 1305 on a rapid thermal processing (RTP) tool 1310. Because the apparatus 1300 is part of an Advanced Process Control (APC) system, the workpieces 1305 are processed on a run-to-run basis. Thus, process adjustments are made and held constant for the duration of a run, based on run-level measurements or averages. A “run” may be a lot, a batch of lots, or even an individual wafer.

In this particular embodiment, the workpieces 1305 are processed by the rapid thermal processing (RTP) tool 1310 and various operations in the process are controlled by a plurality of rapid thermal processing (RTP) control input signals on a line 1320 between the rapid thermal processing (RTP) tool 1310 and a workstation 1330. Exemplary rapid thermal processing (RTP) control inputs for this embodiment might include those for the setpoints for workpiece temperature, lamp power, anneal time, anneal sequence, gas environment, and the like.

When a process step in the rapid thermal processing (RTP) tool 1310 is concluded, the semiconductor workpieces 1305 being processed in the rapid thermal processing (RTP) tool 1310 are examined at a review station 1317. The rapid thermal processing (RTP) control inputs generally affect the characteristic parameters of the semiconductor workpieces 1305 measured at the review station 1317, and, hence, the variability and properties of the acts performed by the rapid thermal processing (RTP) tool 1310 on the workpieces 1305. Once errors are determined from the examination after the run of a lot of workpieces 1305, the rapid thermal processing (RTP) control inputs on the line 1320 are modified for a subsequent run of a lot of workpieces 1305. Modifying the control signals on the line 1320 is designed to improve the next processing performed by the rapid thermal processing (RTP) tool 1310. The modification is performed in accordance with one particular embodiment of the method 1100 set forth in FIG. 11, as described more fully below. Once the relevant rapid thermal processing (RTP) control input signals for the rapid thermal processing (RTP) tool 1310 are updated, the rapid thermal processing (RTP) control input signals with new settings are used for a subsequent run of semiconductor devices.

Referring now to both FIGS. 13 and 14, the rapid thermal processing (RTP) tool 1310 communicates with a manufacturing framework comprising a network of processing modules. One such module is an Advanced Process Control (APC) system manager 1440 resident on the computer 1340. This network of processing modules constitutes the Advanced Process Control (APC) system. The rapid thermal processing (RTP) tool 1310 generally comprises an equipment interface 1410 and a sensor interface 1415. A machine interface 1430 resides on the workstation 1330. The machine interface 1430 bridges the gap between the Advanced Process Control (APC) framework, e.g., the Advanced Process Control (APC) system manager 1440, and the equipment interface 1410. Thus, the machine interface 1430 interfaces the rapid thermal processing (RTP) tool 1310 with the Advanced Process Control (APC) framework and supports machine setup, activation, monitoring, and data collection. The sensor interface 1415 provides the appropriate interface environment to communicate with external sensors such as LabView® or other sensor bus-based data acquisition software. Both the machine interface 1430 and the sensor interface 1415 use a set of functionalities (such as a communication standard) to collect data to be used. The equipment interface 1410 and the sensor interface 1415 communicate over the line 1320 with the machine interface 1430 resident on the workstation 1330.

More particularly, the machine interface 1430 receives commands, status events, and collected data from the equipment interface 1410 and forwards these as needed to other Advanced Process Control (APC) components and event channels. In turn, responses from Advanced Process Control (APC) components are received by the machine interface 1430 and rerouted to the equipment interface 1410. The machine interface 1430 also reformats and restructures messages and data as necessary. The machine interface 1430 supports the startup/shutdown procedures within the Advanced Process Control (APC) System Manager 1440. It also serves as an Advanced Process Control (APC) data collector, buffering data collected by the equipment interface 1410, and emitting appropriate data collection signals.

In the particular embodiment illustrated, the Advanced Process Control (APC) system is a factory-wide software system, but this is not necessary to the practice of the invention. The control strategies taught by the present invention can be applied to virtually any semiconductor rapid thermal processing (RTP) tool on a factory floor. Indeed, the present invention may be simultaneously employed on multiple rapid thermal processing (RTP) tools in the same factory or in the same fabrication process. The Advanced Process Control (APC) framework permits remote access and monitoring of the process performance. Furthermore, by utilizing the Advanced Process Control (APC) framework, data storage can be more convenient, more flexible, and less expensive than data storage on local drives. However, the present invention may be employed, in some alternative embodiments, on local drives.

The illustrated embodiment deploys the present invention onto the Advanced Process Control (APC) framework utilizing a number of software components. In addition to components within the Advanced Process Control (APC) framework, a computer script is written for each of the semiconductor rapid thermal processing (RTP) tools involved in the control system. When a semiconductor rapid thermal processing (RTP) tool in the control system is started in the semiconductor manufacturing fab, the semiconductor rapid thermal processing (RTP) tool generally calls upon a script to initiate the action that is required by the rapid thermal processing (RTP) tool controller. The control methods are generally defined and performed using these scripts. The development of these scripts can comprise a significant portion of the development of a control system.

In this particular embodiment, there are several separate software scripts that perform the tasks involved in controlling the rapid thermal processing (RTP) operation. There is one script for the rapid thermal processing (RTP) tool 1310, including the review station 1317 and the rapid thermal processing (RTP) tool controller 1315. There is also a script to handle the actual data capture from the review station 1317 and another script that contains common procedures that can be referenced by any of the other scripts. There is also a script for the Advanced Process Control (APC) system manager 1440. The precise number of scripts, however, is implementation specific and alternative embodiments may use other numbers of scripts.

Operation of an Illustrative Apparatus. FIG. 15 illustrates one particular embodiment 1500 of the method 1100 in FIG. 11. The method 1500 may be practiced with the apparatus 1300 illustrated in FIGS. 13-14, but the invention is not so limited. The method 1500 may be practiced with any apparatus that may perform the functions set forth in FIG. 15. Furthermore, the method 1100 in FIG. 11 may be practiced in embodiments alternative to the method 1500 in FIG. 15.

Referring now to all of FIGS. 13-15, the method 1500 begins with processing a lot of workpieces 1305 through a rapid thermal processing (RTP) tool, such as the rapid thermal processing (RTP) tool 1310, as set forth in box 1510. In this particular embodiment, the rapid thermal processing (RTP) tool 1310 has been initialized for processing by the Advanced Process Control (APC) system manager 1440 through the machine interface 1430 and the equipment interface 1410. In this particular embodiment, before the rapid thermal processing (RTP) tool 1310 is run, the Advanced Process Control (APC) system manager script is called to initialize the rapid thermal processing (RTP) tool 1310. At this step, the script records the identification number of the rapid thermal processing (RTP) tool 1310 and the lot number of the workpieces 1305. The identification number is then stored against the lot number in a data store 1360. The rest of the script, such as the APCData call and the Setup and StartMachine calls, are formulated with blank or dummy data in order to force the machine to use default settings.

As part of this initialization, the initial setpoints for rapid thermal processing (RTP) control are provided to the rapid thermal processing (RTP) tool controller 1315 over the line 1320. These initial setpoints may be determined and implemented in any suitable manner known to the art. In the particular embodiment illustrated, rapid thermal processing (RTP) controls are implemented by control threads. Each control thread acts like a separate controller and is differentiated by various process conditions. For rapid thermal processing (RTP) control, the control threads are separated by a combination of different conditions. These conditions may include, for example, the semiconductor rapid thermal processing (RTP) tool 1310 currently processing the wafer lot, the semiconductor product, the semiconductor manufacturing operation, and one or more of the semiconductor processing tools (not shown) that previously processed the semiconductor wafer lot.

Control threads are separated because different process conditions affect the rapid thermal processing (RTP) error differently. By isolating each of the process conditions into its own corresponding control thread, the rapid thermal processing (RTP) error can become a more accurate portrayal of the conditions in which a subsequent semiconductor wafer lot in the control thread will be processed. Since the error measurement is more relevant, changes to the rapid thermal processing (RTP) control input signals based upon the error will be more appropriate.

The control thread for the rapid thermal processing (RTP) control scheme depends upon the current rapid thermal processing (RTP) tool, current operation, the product code for the current lot, and the identification number at a previous processing step. The first three parameters are generally found in the context information that is passed to the script from the rapid thermal processing (RTP) tool 1310. The fourth parameter is generally stored when the lot is previously processed. Once all four parameters are defined, they are combined to form the control thread name; RTPP02_OPER01_PROD01_RTPP01 is an example of a control thread name. The control thread name is also stored in correspondence to the wafer lot number in the data store 1360.

Once the lot is associated with a control thread name, the initial settings for that control thread are generally retrieved from the data store 1360. There are at least two possibilities when the call is made for the information. One possibility is that there are no settings stored under the current control thread name. This can happen when the control thread is new, or if the information was lost or deleted. In these cases, the script initializes the control thread assuming that there is no error associated with it and uses the target values of the rapid thermal processing (RTP) as the rapid thermal processing (RTP) control input settings. It is preferred that the controllers use the default machine settings as the initial settings. By assuming some settings, rapid thermal processing (RTP) errors can be related back to the control settings to facilitate feedback control.

Another possibility is that the initial settings are stored under the control thread name. In this case, one or more wafer lots have been processed under the same control thread name as the current wafer lot, and have also been measured for rapid thermal processing (RTP) error using the review station 1317. When this information exists, the rapid thermal processing (RTP) control input signal settings are retrieved from the data store 1360. These settings are then downloaded to the rapid thermal processing (RTP) tool 1310.

The workpieces 1305 are processed through the rapid thermal processing (RTP) tool 1310. This comprises, in the embodiment illustrated, subjecting the workpieces 1305 to a rapid thermal anneal. The workpieces 1305 are measured on the review station 1317 after their rapid thermal processing (RTP) on the rapid thermal processing (RTP) tool 1310. The review station 1317 examines the workpieces 1305 after they are processed for a number of errors. The data generated by the instruments of the review station 1317 is passed to the machine interface 1430 via sensor interface 1415 and the line 1320. The review station script begins with a number of Advanced Process Control (APC) commands for the collection of data. The review station script then locks itself in place and activates a data available script. This script facilitates the actual transfer of the data from the review station 1317 to the Advanced Process Control (APC) framework. Once the transfer is completed, the script exits and unlocks the review station script. The interaction with the review station 1317 is then generally complete.

As will be appreciated by those skilled in the art having the benefit of this disclosure, the data generated by the review station 1317 should be preprocessed for use. Review stations, such as KLA review stations, provide the control algorithms for measuring the control error. Each of the error measurements, in this particular embodiment, corresponds to one of the rapid thermal processing (RTP) control input signals on the line 1320 in a direct manner. Before the error can be utilized to correct the rapid thermal processing (RTP) control input signal, a certain amount of preprocessing is generally completed.

For example, preprocessing may include outlier rejection. Outlier rejection is a gross error check ensuring that the received data is reasonable in light of the historical performance of the process. This procedure involves comparing each of the rapid thermal processing (RTP) errors to its corresponding predetermined boundary parameter. In one embodiment, even if one of the predetermined boundaries is exceeded, the error data from the entire semiconductor wafer lot is generally rejected.

To determine the limits of the outlier rejection, thousands of actual semiconductor manufacturing fabrication (“fab”) data points are collected. The standard deviation for each error parameter in this collection of data is then calculated. In one embodiment, for outlier rejection, nine times the standard deviation (both positive and negative) is generally chosen as the predetermined boundary. This was done primarily to ensure that only the points that are significantly outside the normal operating conditions of the process are rejected.

Preprocessing may also smooth the data, which is also known as filtering. Filtering is important because the error measurements are subject to a certain amount of randomness, such that the error significantly deviates in value. Filtering the review station data results in a more accurate assessment of the error in the rapid thermal processing (RTP) control input signal settings. In one embodiment, the rapid thermal processing (RTP) control scheme utilizes a filtering procedure known as an Exponentially-Weighted Moving Average (“EWMA”) filter, although other filtering procedures can be utilized in this context.

One embodiment for the EWMA filter is represented by Equation (1):

AVG _(N) =W*M _(C)+(1−W)*AVG _(P)  (1)

where

AVG_(N)≡the new EWMA average;

W≡a weight for the new average (AVG_(N));

M_(C)≡the current measurement; and

AVG_(P)≡the previous EWMA average.

The weight is an adjustable parameter that can be used to control the amount of filtering and is generally between zero and one. The weight represents the confidence in the accuracy of the current data point. If the measurement is considered accurate, the weight should be close to one. If there were a significant amount of fluctuations in the process, then a number closer to zero would be appropriate.

In one embodiment, there are at least two techniques for utilizing the EWMA filtering process. The first technique uses the previous average, the weight, and the current measurement as described above. Among the advantages of utilizing the first implementation are ease of use and minimal data storage. One of the disadvantages of utilizing the first implementation is that this method generally does not retain much process information. Furthermore, the previous average calculated in this manner would be made up of every data point that preceded it, which may be undesirable. The second technique retains only some of the data and calculates the average from the raw data each time.

The manufacturing environment in the semiconductor manufacturing fab presents some unique challenges. The order that the semiconductor wafer lots are processed through an rapid thermal processing (RTP) tool may not correspond to the order in which they are read on the review station. This could lead to the data points being added to the EWMA average out of sequence. Semiconductor wafer lots may be analyzed more than once to verify the error measurements. With no data retention, both readings would contribute to the EWMA average, which may be an undesirable characteristic. Furthermore, some of the control threads may have low volume, which may cause the previous average to be outdated such that it may not be able to accurately represent the error in the rapid thermal processing (RTP) control input signal settings.

The rapid thermal processing (RTP) tool controller 1315, in this particular embodiment, uses limited storage of data to calculate the EWMA filtered error, i.e., the first technique. Wafer lot data, including the lot number, the time the lot was processed, and the multiple error estimates, are stored in the data store 1360 under the control thread name. When a new set of data is collected, the stack of data is retrieved from data store 1360 and analyzed. The lot number of the current lot being processed is compared to those in the stack. If the lot number matches any of the data present there, the error measurements are replaced. Otherwise, the data point is added to the current stack in chronological order, according to the time periods when the lots were processed. In one embodiment, any data point within the stack that is over 128 hours old is removed. Once the aforementioned steps are complete, the new filter average is calculated and stored to data store 1360.

Thus, the data is collected and preprocessed, and then processed to generate an estimate of the current errors in the rapid thermal processing (RTP) control input signal settings. First, the data is passed to a compiled Matlab® plug-in that performs the outlier rejection criteria described above. The inputs to a plug-in interface are the multiple error measurements and an array containing boundary values. The return from the plug-in interface is a single toggle variable. A nonzero return denotes that it has failed the rejection criteria, otherwise the variable returns the default value of zero and the script continues to process.

After the outlier rejection is completed, the data is passed to the EWMA filtering procedure. The controller data for the control thread name associated with the lot is retrieved, and all of the relevant operation upon the stack of lot data is carried out. This comprises replacing redundant data or removing older data. Once the data stack is adequately prepared, it is parsed into ascending time-ordered arrays that correspond to the error values. These arrays are fed into the EWMA plug-in along with an array of the parameter required for its execution. In one embodiment, the return from the plug-in is comprised of the six filtered error values.

Returning to FIG. 15, data preprocessing comprises monitoring and/or measuring workpiece 1305 parameter(s) characteristic of the rapid thermal processing (RTP) tool 1310 variables, as set forth in box 1520. Known, potential characteristic parameters may be identified by characteristic data patterns or may be identified as known consequences of modifications to rapid thermal processing (RTP) control. In turn, the rapid thermal processing (RTP) control input parameters such as the rapid thermal processing (RTP) recipe control input parameters and/or the setpoints for workpiece temperature and/or lamp power and/or anneal time and/or ramp rates and/or anneal recipes and/or the gas environment may directly affect the effective yield of usable semiconductor devices from the workpiece 1205.

The next step in the control process is to calculate the new settings for the rapid thermal processing (RTP) tool controller 1315 of the rapid thermal processing (RTP) tool 1310. The previous settings for the control thread corresponding to the current wafer lot are retrieved from the data store 1360. This data is paired along with the current set of rapid thermal processing (RTP) errors. The new settings are calculated by calling a compiled Matlab® plug-in. This application incorporates a number of inputs, performs calculations in a separate execution component, and returns a number of outputs to the main script. Generally, the inputs of the Matlab® plug-in are the rapid thermal processing (RTP) control input signal settings, the review station 1317 errors, an array of parameters that are necessary for the control algorithm, and a currently unused flag error. The outputs of the Matlab® plug-in are the new controller settings, calculated in the plug-in according to the controller algorithm described above.

A rapid thermal processing (RTP) process engineer or a control engineer, who generally determines the actual form and extent of the control action, can set the parameters. They include the threshold values, maximum step sizes, controller weights, and target values. Once the new parameter settings are calculated, the script stores the setting in the data store 1360 such that the rapid thermal processing (RTP) tool 1310 can retrieve them for the next wafer lot to be processed. The principles taught by the present invention can be implemented into other types of manufacturing frameworks.

Returning again to FIG. 15, the calculation of new settings comprises, as set forth in box 1530, modeling the characteristic parameter(s) using a first-principles radiation model. This modeling may be performed by the Matlab® plug-in. In this particular embodiment, only known, potential characteristic parameters are modeled and the models are stored in a database 1335 accessed by a machine interface 1430. The database 1335 may reside on the workstation 1330, as shown, or some other part of the Advanced Process Control (APC) framework. For instance, the models might be stored in the data store 1360 managed by the Advanced Process Control (APC) system manager 1440 in alternative embodiments. The model will generally be a mathematical model, i.e., an equation describing how the change(s) in rapid thermal processing (RTP) recipe control(s) affects the rapid thermal processing (RTP) performance, and the like. The models described in various illustrative embodiments given above, and described more fully below, are examples of such models.

The particular model used will be implementation specific, depending upon the particular rapid thermal processing (RTP) tool 1310 and the particular characteristic parameter(s) being modeled. Whether the relationship in the model is linear or non-linear will be dependent on the particular parameter(s) involved.

The new settings are then transmitted to and applied by the rapid thermal processing (RTP) tool controller 1315. Thus, returning now to FIG. 15, once the characteristic parameter(s) are modeled, the model is applied to modify at least one rapid thermal processing (RTP) recipe control input parameter using at least one proportional-integral-derivative (PID) controller, described more fully above, as set forth in box 1540. In this particular embodiment, the machine interface 1430 retrieves the model from the database 1335, plugs in the respective value(s), and determines the necessary change(s) in the rapid thermal processing (RTP) recipe control input parameter(s). The change is then communicated by the machine interface 1430 to the equipment interface 1410 over the line 1320. The equipment interface 1410 then implements the change.

The present embodiment furthermore provides that the models be updated. This comprises, as set forth in boxes 1550-1560 of FIG. 15, monitoring at least one effect of modifying the rapid thermal processing (RTP) recipe control input parameters (box 1550) and updating the applied model (box 1560) based on the effect(s) monitored. For instance, various aspects of the operation of the rapid thermal processing (RTP) tool 1310 will change as the rapid thermal processing (RTP) tool 1310 ages. By monitoring the effect of the rapid thermal processing (RTP) recipe change(s) implemented as a result of the characteristic parameter measurement, the necessary value could be updated to yield superior performance.

As noted above, this particular embodiment implements an Advanced Process Control (APC) system. Thus, changes are implemented “between” workpiece lots. The actions set forth in the boxes 1520-1560 are implemented after the current workpiece lot is processed and before the second workpiece lot is processed, as set forth in box 1570 of FIG. 15. However, the invention is not so limited. For example, in alternative illustrative embodiments, changes may be implemented “between” individual workpieces. Furthermore, as noted above, a lot may constitute any practicable number of wafers from one to several thousand (or practically any finite number). What constitutes a “lot” is implementation specific, and so the point of the fabrication process in which the updates occur will vary from implementation to implementation.

As described above, in various illustrative embodiments of the present invention, a first-principles radiation model may be applied to modify rapid thermal processing performed in a rapid thermal processing step. For example, a first-principles radiation model may be formed by monitoring one or more tool variables and/or one or more rapid thermal processing parameters during one or more rapid thermal processing runs. Examples of such tool variables and/or rapid thermal processing parameters may comprise one or more pyrometer trace readings, one or more lamp power trace readings, one or more tube temperature trace readings, gas composition, gas flow rates, pressure, and the like.

In mathematical terms, a set of m rapid thermal processing runs being measured and/or monitored over n rapid thermal processing tool variables and/or rapid thermal processing parameters may be arranged as a rectangular n×m matrix X. In other words, the rectangular n×m matrix X may be comprised of 1 to n rows (each row corresponding to a separate rapid thermal processing tool variable or rapid thermal processing parameter) and 1 to m columns (each column corresponding to a separate rapid thermal processing run). The values of the rectangular n×m matrix X may be the actually measured values of the rapid thermal processing tool variables and/or rapid thermal processing parameters, or ratios of actually measured values (normalized to respective reference setpoints), or logarithms of such ratios, for example. The rectangular n×m matrix X may have rank r, where r≦min{m,n} is the maximum number of independent variables in the matrix X. The rectangular n×m matrix X may be analyzed using Principle Components Analysis (PCA), for example. The use of PCA, for example, generates a set of Principal Components P (whose “Loadings,” or components, represent the contributions of the various rapid thermal processing tool variables and/or rapid thermal processing parameters) as an eigenmatrix (a matrix whose columns are eigenvectors) of the equation ((X−M)(X−M)^(T))P=Λ²P, where M is a rectangular n×m matrix of the mean values of the columns of X (the m columns of M are each the column mean vector μ_(n×1) of X_(n×m)), Λ² is an n×n diagonal matrix of the squares of the eigenvalues λ_(i), i=1,2, . . . ,r, of the mean-scaled matrix X−M, and a Scores matrix, T, with X−M=PT^(T) and (X−M)^(T)=(PT^(T))^(T)=(T^(T))^(T)P^(T)=TP^(T), so that ((X−M)(X−M)^(T))P=((PT^(T))(TP^(T)))P and ((PT^(T))(TP^(T)))P=(P(T^(T)T)P^(T))P=P(T^(T)T)=Λ²P. The rectangular n×m matrix X, also denoted X_(n×m), may have elements x_(ij), where i=1,2, . . . ,n, and j=1,2, . . . ,m, and the rectangular m×n matrix X^(T), the transpose of the rectangular n×m matrix X, also denoted (X^(T))_(m×n), may have elements x_(ji), where i=1,2, . . . ,n, and j=1,2, . . . ,m. The n×n matrix (X−M)(X−M)^(T) is (m−1) times the covariance matrix S_(n×n), having elements s_(ij), where i=1,2, . . . ,n, and j=1,2, . . . , n, defined so that: ${s_{ij} = \frac{{m{\sum\limits_{k = 1}^{m}{x_{lk}x_{jk}}}} - {\sum\limits_{k = 1}^{m}{x_{tk}{\sum\limits_{k = 1}^{m}x_{jk}}}}}{m\left( {m - 1} \right)}},$

corresponding to the rectangular n×m matrix X_(n×m).

Although other methods may exist, four methods for computing Principal Components are as follows:

1. eigenanalysis (EIG);

2. singular value decomposition (SVD);

3. nonlinear iterative partial least squares (NIPALS); and

4. power method.

Each of the first two methods, EIG and SVD, simultaneously calculates all possible Principal Components, whereas the NIPALS method allows for calculation of one Principal Component at a time. However, the power method, described more fully below, is an iterative approach to finding eigenvalues and eigenvectors, and also allows for calculation of one Principal Component at a time. There are as many Principal Components as there are channels (or wavelengths or frequencies). The power method may efficiently use computing time.

For example, consider the 3×2 matrix A, its transpose, the 2×3 matrix A^(T), their 2×2 matrix product A^(T)A, and their 3×3 matrix product AA^(T): $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ 1 & {- 1} \end{pmatrix}$ $A^{T} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & {- 1} \end{pmatrix}$ ${A^{T}A} = {{\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & {- 1} \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & 0 \\ 1 & {- 1} \end{pmatrix}} = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}}$ ${AA}^{T} = {{\begin{pmatrix} 1 & 1 \\ 1 & 0 \\ 1 & {- 1} \end{pmatrix}\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & {- 1} \end{pmatrix}} = {\begin{pmatrix} 2 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 2 \end{pmatrix}.}}$

EIG reveals that the eigenvalues λ of the matrix product A^(T)A are 3 and 2. The eigenvectors of the matrix product A^(T)A are solutions t of the equation (A^(T)A)t=λt, and may be seen by inspection to be t₁ ^(T)=(1,0) and t₂ ^(T)=(0,1), belonging to the eigenvalues λ₁=3 and λ₂=2, respectively.

The power method, for example, may be used to determine the eigenvalues λ and eigenvectors p of the matrix product AA^(T), where the eigenvalues λ and the eigenvectors p are solutions p of the equation (AA^(T))p=λp. A trial eigenvector p^(T)=(1,1,1) may be used: ${\left( {AA}^{T} \right)\underset{\_}{p}} = {{\begin{pmatrix} 2 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}} = {\begin{pmatrix} 3 \\ 3 \\ 3 \end{pmatrix} = {{3\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}} = {\lambda_{1}{{\underset{\_}{p}}_{1}.}}}}}$

This indicates that the trial eigenvector p^(T)=(1,1,1) happened to correspond to the eigenvector p₁ ^(T)=(1,1,1) belonging to the eigenvalue λ₁=3. The power method then proceeds by subtracting the outer product matrix p₁p₁ ^(T) from the matrix product AA^(T) to form a residual matrix R₁: $R_{1} = {{\begin{pmatrix} 2 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 2 \end{pmatrix} - {\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\begin{pmatrix} 1 & 1 & 1 \end{pmatrix}}} = {{\begin{pmatrix} 2 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 2 \end{pmatrix} - \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}} = {\begin{pmatrix} 1 & 0 & {- 1} \\ 0 & 0 & 0 \\ {- 1} & 0 & 1 \end{pmatrix}.}}}$

Another trial eigenvector p^(T)=(1,0,−1) may be used: ${\left( {{AA}^{T} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}}} \right)\underset{\_}{p}} = {{R_{1}\underset{\_}{p}} = {{\begin{pmatrix} 1 & 0 & {- 1} \\ 0 & 0 & 0 \\ {- 1} & 0 & 1 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \\ {- 1} \end{pmatrix}} = {\begin{pmatrix} 2 \\ 0 \\ {- 2} \end{pmatrix} = {{2\begin{pmatrix} 1 \\ 0 \\ {- 1} \end{pmatrix}} = {\lambda_{2}{{\underset{\_}{p}}_{2}.}}}}}}$

This indicates that the trial eigenvector p^(T)=(1,0,−1) happened to correspond to the eigenvector p₂ ^(T)=(1,0,−1) belonging to the eigenvalue λ₂=2. The power method then proceeds by subtracting the outer product matrix p₂p₂ ^(T) from the residual matrix R₁ to form a second residual matrix R₂: $R_{2} = {{\begin{pmatrix} 1 & 0 & {- 1} \\ 0 & 0 & 0 \\ {- 1} & 0 & 1 \end{pmatrix} - {\begin{pmatrix} 1 \\ 0 \\ {- 1} \end{pmatrix}\begin{pmatrix} 1 & 0 & {- 1} \end{pmatrix}}} = {{\begin{pmatrix} 1 & 0 & {- 1} \\ 0 & 0 & 0 \\ {- 1} & 0 & 1 \end{pmatrix} - \begin{pmatrix} 1 & 0 & {- 1} \\ 0 & 0 & 0 \\ {- 1} & 0 & 1 \end{pmatrix}} = {\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}.}}}$

The fact that the second residual matrix R₂ vanishes indicates that the eigenvalue λ₃=0 and that the eigenvector p₃ is completely arbitrary. The eigenvector p₃ may be conveniently chosen to be orthogonal to the eigenvectors p₁ ^(T)=(1,1,1) and p₂ ^(T)=(1,0,−1), so that the eigenvector p₃ ^(T)=(1,−2,1). Indeed, one may readily verify that: ${\left( {AA}^{T} \right){\underset{\_}{p}}_{3}} = {{\begin{pmatrix} 2 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ {- 2} \\ 1 \end{pmatrix}} = {\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} = {{0\begin{pmatrix} 1 \\ {- 2} \\ 1 \end{pmatrix}} = {\lambda_{3}{{\underset{\_}{p}}_{3}.}}}}}$

Similarly, SVD of A shows that A=PT^(T), where P is the Principal Component matrix and T is the Scores matrix: $A = {{\begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & 0 & \frac{- 2}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & \frac{- 1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \end{pmatrix}\begin{pmatrix} \sqrt{3} & 0 \\ 0 & \sqrt{2} \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}} = {\begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & 0 & \frac{- 2}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & \frac{- 1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \end{pmatrix}{\begin{pmatrix} \sqrt{3} & 0 \\ 0 & \sqrt{2} \\ 0 & 0 \end{pmatrix}.}}}$

SVD confirms that the singular values of A are 3 and 2, the positive square roots of the eigenvalues λ₁=3 and λ₂=2 of the matrix product A^(T)A. Note that the columns of the Principal Component matrix P are the orthonormalized eigenvectors of the matrix product AA^(T).

Likewise, SVD of A^(T) shows that A^(T)=TP^(T): $A^{T} = {{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{2} & 0 \end{pmatrix}\begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{- 1}{\sqrt{2}} \\ \frac{1}{\sqrt{6}} & \frac{- 2}{\sqrt{6}} & \frac{1}{\sqrt{6}} \end{pmatrix}} = {A^{T} = {{\begin{pmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{2} & 0 \end{pmatrix}\begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{- 1}{\sqrt{2}} \\ \frac{1}{\sqrt{6}} & \frac{- 2}{\sqrt{6}} & \frac{1}{\sqrt{6}} \end{pmatrix}} = {{TP}^{T}.}}}}$

SVD confirms that the (non-zero) singular values of A^(T) are 3 and 2, the positive square roots of the eigenvalues λ₁=3 and λ₂=2 of the matrix product AA^(T). Note that the columns of the Principal Component matrix P (the rows of the Principal Component matrix P^(T)) are the orthonormalized eigenvectors of the matrix product AA^(T). Also note that the non-zero elements of the Scores matrix T are the positive square roots 3 and 2 of the (non-zero) eigenvalues λ₁=3 and λ₂=2 of both of the matrix products A^(T)A and AA^(T).

Taking another example, consider the 4×3 matrix B, its transpose, the 3×4 matrix B^(T), their 3×3 matrix product B^(T)B, and their 4×4 matrix product BB^(T): $B = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & {- 1} \\ 1 & {- 1} & 0 \end{pmatrix}$ $B^{T} = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \end{pmatrix}$ ${B^{T}B} = {{\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \end{pmatrix}\begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & {- 1} \\ 1 & {- 1} & 0 \end{pmatrix}} = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}}$ ${BB}^{T} = {{\begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & {- 1} \\ 1 & {- 1} & 0 \end{pmatrix}\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \end{pmatrix}} = {\begin{pmatrix} 2 & 1 & 1 & 0 \\ 1 & 2 & 0 & 1 \\ 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \end{pmatrix}.}}$

EIG reveals that the eigenvalues of the matrix product B^(T)B are 4, 2 and 2. The eigenvectors of the matrix product B^(T)B are solutions t of the equation (B^(T)B)t=λt, and may be seen by inspection to be t₁ ^(T)=(1,0,0), t₂ ^(T)=(0,1,0), and t₃ ^(T)=(0,0,1), belonging to the eigenvalues λ₁=4, λ₂=2, and λ₃=2, respectively.

The power method, for example, may be used to determine the eigenvalues λ and eigenvectors p of the matrix product BB^(T), where the eigenvalues λ and the eigenvectors p are solutions p of the equation (BB^(T))p=λp. A trial eigenvector p^(T)=(1,1,1,1) may be used: ${\left( {BB}^{T} \right)\underset{\_}{p}} = {{\begin{pmatrix} 2 & 1 & 1 & 0 \\ 1 & 2 & 0 & 1 \\ 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}} = {\begin{pmatrix} 4 \\ 4 \\ 4 \\ 4 \end{pmatrix} = {{4\begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}} = {\lambda_{1}{{\underset{\_}{p}}_{1}.}}}}}$

This indicates that the trial eigenvector p^(T)=(1,1,1,1) happened to correspond to the eigenvector p₁ ^(T)=(1,1,1,1) belonging to the eigenvalue λ₁=4. The power method then proceeds by subtracting the outer product matrix p₁p₁ ^(T) from the matrix product BB^(T) to form a residual matrix R₁: $R_{1} = {{\begin{pmatrix} 2 & 1 & 1 & 0 \\ 1 & 2 & 0 & 1 \\ 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \end{pmatrix} - \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{pmatrix}} = {\begin{pmatrix} 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \\ 0 & {- 1} & 1 & 0 \\ {- 1} & 0 & 0 & 1 \end{pmatrix}.}}$

Another trial eigenvector p^(T)=(1,0,0,−1) may be used: ${\left( {{BB}^{T} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}}} \right)\underset{\_}{p}} = {{R_{1}\underset{\_}{p}} = {{\begin{pmatrix} 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \\ 0 & {- 1} & 1 & 0 \\ {- 1} & 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \\ 0 \\ {- 1} \end{pmatrix}} = {\begin{pmatrix} 2 \\ 0 \\ 0 \\ {- 2} \end{pmatrix} = {{2\begin{pmatrix} 1 \\ 0 \\ 0 \\ {- 1} \end{pmatrix}} = {\lambda_{2}{{\underset{\_}{p}}_{2}.}}}}}}$

This indicates that the trial eigenvector p^(T)=(1,0,0,−1) happened to correspond to the eigenvector p₂ ^(T)=(1,0,0,−1) belonging to the eigenvalue λ₂=2. The power method then proceeds by subtracting the outer product matrix p₂p₂ ^(T) from the residual matrix R₁ to form a second residual matrix R₂: $R_{2} = {{\begin{pmatrix} 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \\ 0 & {- 1} & 1 & 0 \\ {- 1} & 0 & 0 & 1 \end{pmatrix} - \begin{pmatrix} 1 & 0 & 0 & {- 1} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ {- 1} & 0 & 0 & 1 \end{pmatrix}} = {\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & {- 1} & 0 \\ 0 & {- 1} & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}.}}$

Another trial eigenvector p^(T)=(0,1,−1,0) may be used: ${\left( {{BB}^{T} - {{\underset{\_}{p}}_{2}{\underset{\_}{p}}_{2}^{T}}} \right)\underset{\_}{p}} = {{R_{2}\underset{\_}{p}} = {{\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & {- 1} & 0 \\ 0 & {- 1} & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} 0 \\ 1 \\ {- 1} \\ 0 \end{pmatrix}} = {\begin{pmatrix} 0 \\ 2 \\ {- 2} \\ 0 \end{pmatrix} = {{2\begin{pmatrix} 0 \\ 1 \\ {- 1} \\ 0 \end{pmatrix}} = {\lambda_{3}{{\underset{\_}{p}}_{3}.}}}}}}$

This indicates that the trial eigenvector p^(T)=(0,1,−1,0) happened to correspond to the eigenvector p₃ ^(T)=(0,1,−1,0) belonging to the eigenvalue λ₃=2. The power method then proceeds by subtracting the outer product matrix p₃p₃ ^(T) from the second residual matrix R₂ to form a third residual matrix R₃: $R_{3} = {{\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & {- 1} & 0 \\ 0 & {- 1} & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} - \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & {- 1} & 0 \\ 0 & {- 1} & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}} = {\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}.}}$

The fact that the third residual matrix R₃ vanishes indicates that the eigenvalue λ₄=0 and that the eigenvector p₄ is completely arbitrary. The eigenvector p₄ may be conveniently chosen to be orthogonal to the eigenvectors p₁ ^(T)=(1,1,1,1), p₂ ^(T)=(1,0,0,−1), and p₃ ^(T)=(0,1,−1,0), so that the eigenvector p₄ ^(T)=(1,−1,−1,1). Indeed, one may readily verify that: ${\left( {BB}^{T} \right){\underset{\_}{p}}_{4}} = {{\begin{pmatrix} 2 & 1 & 1 & 0 \\ 1 & 2 & 0 & 1 \\ 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ {- 1} \\ {- 1} \\ 1 \end{pmatrix}} = {\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} = {{0\begin{pmatrix} 1 \\ {- 1} \\ {- 1} \\ 1 \end{pmatrix}} = {\lambda_{4}{{\underset{\_}{p}}_{4}.}}}}}$

In this case, since the eigenvalues λ₂=2 and λ₃=2 are equal, and, hence, degenerate, the eigenvectors p₂ ^(T)=(1,0,0,−1) and p₃ ^(T)=(0,1,−1,0) belonging to the degenerate eigenvalues λ₂=2=λ₃ may be conveniently chosen to be orthonormal. A Gram-Schmidt orthonormalization procedure may be used, for example.

Similarly, SVD of B shows that B=PT^(T), where P is the Principal Component matrix and T is the Scores matrix: $B = {{\begin{pmatrix} \frac{1}{2} & \frac{1}{\sqrt{2}} & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & \frac{- 1}{2} \\ \frac{1}{2} & 0 & \frac{- 1}{\sqrt{2}} & \frac{- 1}{2} \\ \frac{1}{2} & \frac{- 1}{\sqrt{2}} & 0 & \frac{1}{2} \end{pmatrix}\begin{pmatrix} 2 & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}} = {B = {{\begin{pmatrix} \frac{1}{2} & \frac{1}{\sqrt{2}} & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & \frac{- 1}{2} \\ \frac{1}{2} & 0 & \frac{- 1}{\sqrt{2}} & \frac{- 1}{2} \\ \frac{1}{2} & \frac{- 1}{\sqrt{2}} & 0 & \frac{1}{2} \end{pmatrix}\begin{pmatrix} 2 & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 & 0 \end{pmatrix}} = {{PT}^{T}.}}}}$

SVD confirms that the singular values of B are 2, 2 and 2, the positive square roots of the eigenvalues λ₁=4, λ₂=2 and λ₃=2 of the matrix product B^(T)B.

Likewise, SVD of B^(T) shows that: $B^{T} = {{\left( \quad \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\quad \right)\left( \quad \begin{matrix} 2 & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & 0 \end{matrix}\quad \right)\left( \quad \begin{matrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{\sqrt{2}} & 0 & 0 & \frac{- 1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{- 1}{\sqrt{2}} & 0 \\ \frac{1}{2} & \frac{- 1}{2} & \frac{- 1}{2} & \frac{1}{2} \end{matrix}\quad \right)} = {B^{T} = {{\left( \quad \begin{matrix} 2 & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & 0 \end{matrix}\quad \right)\left( \quad \begin{matrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{\sqrt{2}} & 0 & 0 & \frac{- 1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{- 1}{\sqrt{2}} & 0 \\ \frac{1}{2} & \frac{- 1}{2} & \frac{- 1}{2} & \frac{1}{2} \end{matrix}\quad \right)} = {{TP}^{T}.}}}}$

SVD confirms that the (non-zero) singular values of B^(T) are 2, 2, and 2, the positive square roots of the eigenvalues λ₁=4, λ₂=2 and λ₃=2 of the matrix product AA^(T). Note that the columns of the Principal Component matrix P (the rows of the Principal Component matrix P^(T)) are the orthonormalized eigenvectors of the matrix product BB^(T). Also note that the non-zero elements of the Scores matrix T are the positive square roots 2, 2, and 2 of the (non-zero) eigenvalues λ₁=4, λ₂=2 and λ₃=2 of both of the matrix products B^(T)B and BB^(T).

The matrices A and B discussed above have been used for the sake of simplifying the presentation of PCA and the power method, and are much smaller than the data matrices encountered in illustrative embodiments of the present invention. For example, in various illustrative embodiments, about m=20-100 rapid thermal processing runs may be measured and/or monitored over about n=4-100 rapid thermal processing tool variables and/or rapid thermal processing parameters. Brute force modeling, regressing all m=20-100 runs over n=4-100 variables, may constitute an ill-conditioned regression problem. Techniques such as PCA and/or partial least squares (PLS, also known as projection to latent structures) reduce the complexity in such cases by revealing the hierarchical ordering of the data based on levels of decreasing variability. In PCA, this involves finding successive Principal Components. In PLS techniques such as NIPALS, this involves finding successive latent vectors.

As shown in FIG. 16, a scatterplot 1600 of data points 1610 may be plotted in an n-dimensional variable space (n=3 in FIG. 16). The mean vector 1620 may lie at the center of a p-dimensional Principal Component ellipsoid 1630 (p=2 in FIG. 16). The mean vector 1620 may be determined by taking the average of the columns of the overall data matrix X. The Principal Component ellipsoid 1630 may have a first Principal Component 1640 (major axis in FIG. 16), with a length equal to the largest eigenvalue of the mean-scaled data matrix X−M, and a second Principal Component 1650 (minor axis in FIG. 16), with a length equal to the next largest eigenvalue of the mean-scaled data matrix X−M.

For example, the 3×4 matrix B^(T) given above may be taken as the overall data matrix X (again for the sake of simplicity), corresponding to 4 runs over 3 variables. As shown in FIG. 17, a scatterplot 1700 of data points 1710 may be plotted in a 3-dimensional variable space. The mean vector 1720 μ may lie at the center of a 2-dimensional Principal Component ellipsoid 1730 (really a circle, a degenerate ellipsoid). The mean vector 1720 μ may be determined by taking the average of the columns of the overall 3×4 data matrix B^(T). The Principal Component ellipsoid 1730 may have a first Principal Component 1740 (“major” axis in FIG. 17) and a second Principal Component 1750 (“minor” axis in FIG. 17). Here, the eigenvalues of the mean-scaled data matrix B^(T)-M are equal and degenerate, so the lengths of the “major” and “minor” axes in FIG. 17 are equal. As shown in FIG. 17, the mean vector 1720 μ is given by: ${\underset{\_}{\mu} = {{\frac{1}{4}\left\lbrack {\left( \quad \begin{matrix} 1 \\ 1 \\ 0 \end{matrix}\quad \right) + \left( \quad \begin{matrix} 1 \\ 0 \\ 1 \end{matrix}\quad \right) + \left( \quad \begin{matrix} 1 \\ 0 \\ {- 1} \end{matrix}\quad \right) + \left( \quad \begin{matrix} 1 \\ {- 1} \\ 0 \end{matrix}\quad \right)} \right\rbrack} = \left( \quad \begin{matrix} 1 \\ 0 \\ 0 \end{matrix}\quad \right)}},$

and the matrix M has the mean vector 1720 μ for all 4 columns.

Principal Components Analysis (PCA) may be illustrated geometrically. For example, the 3×2 matrix C (similar to the 3×2 matrix A given above): $C = \left( \quad \begin{matrix} 1 & {- 1} \\ 1 & 0 \\ 1 & 1 \end{matrix}\quad \right)$

may be taken as the overall data matrix X (again for the sake of simplicity), corresponding to 2 runs over 3 variables. As shown in FIG. 18, a scatterplot 1800 of data points 1810 and 1820, with coordinates (1,1,1) and (−1,0,1), respectively, may be plotted in a 3-dimensional variable space where the variables are respective rapid thermal processing tool and/or parameter values for each of the 3 variables. The mean vector 1830 μ may lie at the center of a 1-dimensional Principal Component ellipsoid 1840 (really a line, a very degenerate ellipsoid). The mean vector 1830 μ may be determined by taking the average of the columns of the overall 3×2 data matrix C. The Principal Component ellipsoid 1840 may have a first Principal Component 1850 (the “major” axis in FIG. 18, with length 5, lying along a first Principal Component axis 1860) and no second or third Principal Component lying along second or third Principal Component axes 1870 and 1880, respectively. Here, two of the eigenvalues of the mean-scaled data matrix C−M are equal to zero, so the lengths of the “minor” axes in FIG. 18 are both equal to zero. As shown in FIG. 18, the mean vector 1830 μ is given by: ${\underset{\_}{\mu} = {{\frac{1}{2}\left\lbrack {\left( \quad \begin{matrix} 1 \\ 1 \\ 1 \end{matrix}\quad \right) + \left( \quad \begin{matrix} {- 1} \\ 0 \\ 1 \end{matrix}\quad \right)} \right\rbrack} = \left( \quad \begin{matrix} 0 \\ \frac{1}{2} \\ 1 \end{matrix}\quad \right)}},$

and the matrix M has the mean vector 1830 μ for both columns. As shown in FIG. 18, PCA is nothing more than a principal axis rotation of the original variable axes (here, the respective rapid thermal processing tool and/or parameter values for each of the 3 variables) about the endpoint of the mean vector 1830 μ, with coordinates (0,1/2,1) with respect to the original coordinate axes and coordinates [0,0,0] with respect to the new Principal Component axes 1860, 1870 and 1880. The Loadings are merely the direction cosines of the new Principal Component axes 1860, 1870 and 1880 with respect to the original variable axes. The Scores are simply the coordinates of the data points 1810 and 1820, [5^(0.5)/2,0,0] and [−5^(0.5)/2,0,0], respectively, referred to the new Principal Component axes 1860, 1870 and 1880.

The mean-scaled 3×2 data matrix C−M, its transpose, the 2×3 matrix (C−M)^(T), their 2×2 matrix product (C−M)^(T)(C−M), and their 3×3 matrix product (C−M)(C−M)^(T) are given by: ${C - M} = {{\left( \quad \begin{matrix} 1 & {- 1} \\ 1 & 0 \\ 1 & 1 \end{matrix}\quad \right) - \left( \quad \begin{matrix} 0 & 0 \\ \frac{1}{2} & \frac{1}{2} \\ 1 & 1 \end{matrix}\quad \right)} = \left( \quad \begin{matrix} 1 & {- 1} \\ \frac{1}{2} & \frac{- 1}{2} \\ 0 & 0 \end{matrix}\quad \right)}$ $\left( {C - M} \right)^{T} = \begin{pmatrix} 1 & \frac{1}{2} & 0 \\ {- 1} & \frac{- 1}{2} & 0 \end{pmatrix}$ $\begin{matrix} {{\left( {C - M} \right)^{T}\left( {C - M} \right)} = \left( \quad \begin{matrix} 1 & \frac{1}{2} & 0 \\ {- 1} & \frac{- 1}{2} & 0 \end{matrix}\quad \right)} \\ {= \left( \quad \begin{matrix} \frac{5}{4} & \frac{- 5}{4} \\ \frac{- 5}{4} & \frac{5}{4} \end{matrix}\quad \right)} \end{matrix}\left( \quad \begin{matrix} 1 & {- 1} \\ \frac{1}{2} & \frac{- 1}{2} \\ 0 & 0 \end{matrix}\quad \right)$ $\begin{matrix} {{\left( {C - M} \right)\left( {C - M} \right)^{T}} = {\left( \quad \begin{matrix} 1 & {- 1} \\ \frac{1}{2} & \frac{- 1}{2} \\ 0 & 0 \end{matrix}\quad \right)\left( \quad \begin{matrix} 1 & \frac{1}{2} & 0 \\ {- 1} & \frac{- 1}{2} & 0 \end{matrix}\quad \right)}} \\ {= \left( \quad \begin{matrix} 2 & 1 & 0 \\ 1 & \frac{1}{2} & 0 \\ 0 & 0 & 0 \end{matrix}\quad \right)} \end{matrix}$

The 3×3 matrix (C−M)(C−M)^(T) is the covariance matrix S_(3×3), having elements s_(ij), where i=1,2,3, and j=1,2,3, defined so that: ${s_{ij} = \frac{{2{\sum\limits_{k = 1}^{2}{c_{ik}c_{jk}}}} - {\sum\limits_{k = 1}^{2}{c_{ik}{\sum\limits_{k = 1}^{2}c_{jk}}}}}{2\left( {2 - 1} \right)}},$

corresponding to the rectangular 3×2 matrix C_(3×2).

EIG reveals that the eigenvalues λ of the matrix product (C−M)^(T)(C−M) are 5/2 and 0, for example, by finding solutions to the secular equation: ${\begin{matrix} {\frac{5}{4} - \lambda} & \frac{- 5}{4} \\ \frac{- 5}{4} & {\frac{5}{4} - \lambda} \end{matrix}} = 0.$

The eigenvectors of the matrix product (C−M)^(T)(C−M) are solutions t of the equation (C−M)^(T)(C−M)t=λt, which may be rewritten as ((C−M)^(T)(C−M)−λ)t=0. For the eigenvalue λ₁=5/2, the eigenvector t₁ may be seen by ${\begin{pmatrix} {\frac{5}{4} - \lambda} & \frac{- 5}{4} \\ \frac{- 5}{4} & {\frac{5}{4} - \lambda} \end{pmatrix}\underset{\_}{t}} = {{\begin{pmatrix} \frac{- 5}{4} & \frac{- 5}{4} \\ \frac{- 5}{4} & \frac{- 5}{4} \end{pmatrix}\underset{\_}{t}} = 0}$

to be t₁ ^(T)=(1,−1). For the eigenvalue λ₁=0, the eigenvector t₂ may be seen by ${\begin{pmatrix} {\frac{5}{4} - \lambda} & \frac{- 5}{4} \\ \frac{- 5}{4} & {\frac{5}{4} - \lambda} \end{pmatrix}\underset{\_}{t}} = {{\begin{pmatrix} \frac{5}{4} & \frac{- 5}{4} \\ \frac{- 5}{4} & \frac{5}{4} \end{pmatrix}\underset{\_}{t}} = {{0\quad {to}\quad {be}\quad {\underset{\_}{t}}_{2}^{T}} = {\left( {1,1} \right).}}}$

The power method, for example, may be used to determine the eigenvalues λ and eigenvectors p of the matrix product (C−M)(C−M)^(T), where the eigenvalues λ and the eigenvectors p are solutions p of the equation ((C−M)(C−M)^(T))p=λp. A trial eigenvector p^(T)=(1,1,1) may be used: $\begin{matrix} {{\left( {\left( {C - M} \right)\left( {C - M} \right)^{T}} \right)\underset{\_}{p}} = {\begin{pmatrix} 2 & 1 & 0 \\ 1 & \frac{1}{2} & 0 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}}} \\ {= {\begin{pmatrix} 3 \\ \frac{3}{2} \\ 0 \end{pmatrix} = {{3\begin{pmatrix} 1 \\ \frac{1}{2} \\ 0 \end{pmatrix}} = {3\underset{\_}{q}}}}} \end{matrix}$ $\begin{matrix} {{\left( {\left( {C - M} \right)\left( {C - M} \right)^{T}} \right)\underset{\_}{q}} = {\begin{pmatrix} 2 & 1 & 0 \\ 1 & \frac{1}{2} & 0 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ \frac{1}{2} \\ 0 \end{pmatrix}}} \\ {= {\begin{pmatrix} \frac{5}{2} \\ \frac{5}{4} \\ 0 \end{pmatrix} = {{\frac{5}{2}\begin{pmatrix} 1 \\ \frac{1}{2} \\ 0 \end{pmatrix}} = {\lambda_{1}{{\underset{\_}{p}}_{1}.}}}}} \end{matrix}$

This illustrates that the trial eigenvector p^(T)=(1,1,1) gets replaced by the improved trial eigenvector q^(T)=(1,1/2,0) that happened to correspond to the eigenvector p₁ ^(T)=(1,1/2,0) belonging to the eigenvalue λ₁=5/2. The power method then proceeds by subtracting the outer product matrix p₁p₁ ^(T) from the matrix product (C−M)(C−M)^(T) to form a residual matrix R₁: $\begin{matrix} {R_{1} = {\begin{pmatrix} 2 & 1 & 0 \\ 1 & \frac{1}{2} & 0 \\ 0 & 0 & 0 \end{pmatrix} - {\begin{pmatrix} 1 \\ \frac{1}{2} \\ 0 \end{pmatrix}\quad \begin{pmatrix} 1 & \frac{1}{2} & 0 \end{pmatrix}}}} \\ {= {\begin{pmatrix} 2 & 1 & 0 \\ 1 & \frac{1}{2} & 0 \\ 0 & 0 & 0 \end{pmatrix} - \begin{pmatrix} 1 & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{4} & 0 \\ 0 & 0 & 0 \end{pmatrix}}} \end{matrix}$ $R_{1} = {\begin{pmatrix} 1 & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{4} & 0 \\ 0 & 0 & 0 \end{pmatrix}.}$

Another trial eigenvector p^(T)=(−1,2,0), orthogonal to the eigenvector p₁ ^(T)=(1,1/2,0) may be used: $\begin{matrix} {{\left( {{\left( {C - M} \right)\quad \left( {C - M} \right)^{T}} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}}} \right)\underset{\_}{p}} = {R_{1}p}} \\ {= {\begin{pmatrix} 1 & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{4} & 0 \\ 0 & 0 & 0 \end{pmatrix}\quad \begin{pmatrix} {- 1} \\ 2 \\ 0 \end{pmatrix}}} \\ {= {\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} = {{0\quad \begin{pmatrix} {- 1} \\ 2 \\ 0 \end{pmatrix}} = {\lambda_{2}{{\underset{\_}{p}}_{2}.}}}}} \end{matrix}$

This indicates that the trial eigenvector p^(T)=(−1,2,0) happened to correspond to the eigenvector p₂ ^(T)=(−1,2,0) belonging to the eigenvalue λ₂=0. The power method then proceeds by subtracting the outer product matrix p₂p₂ ^(T) from the residual matrix R₁ to form a second residual matrix R₂: $\begin{matrix} {R_{2} = {\begin{pmatrix} 1 & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{4} & 0 \\ 0 & 0 & 0 \end{pmatrix} - {\begin{pmatrix} {- 1} \\ 2 \\ 0 \end{pmatrix}\quad \begin{pmatrix} {- 1} & 2 & 0 \end{pmatrix}}}} \\ {= {\begin{pmatrix} 1 & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{4} & 0 \\ 0 & 0 & 0 \end{pmatrix} - \begin{pmatrix} 1 & {- 2} & 0 \\ {- 2} & 4 & 0 \\ 0 & 0 & 0 \end{pmatrix}}} \end{matrix}$ $R_{2} = {\begin{pmatrix} 0 & \frac{5}{2} & 0 \\ \frac{5}{2} & \frac{- 15}{4} & 0 \\ 0 & 0 & 0 \end{pmatrix}.}$

Another trial eigenvector p^(T)=(0,0,1), orthogonal to the eigenvectors p₁ ^(T)=(1,1/2,0) and p₂ ^(T)=(−1,2,0) may be used: $\begin{matrix} {{\left( {{\left( {C - M} \right)\quad \left( {C - M} \right)^{T}} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}} - {{\underset{\_}{p}}_{2}{\underset{\_}{p}}_{2}^{T}}} \right)\underset{\_}{p}} = {R_{2}\underset{\_}{p}}} \\ {= {\begin{pmatrix} 1 & \frac{5}{2} & 0 \\ \frac{5}{2} & \frac{- 15}{4} & 0 \\ 0 & 0 & 0 \end{pmatrix}\quad \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}}} \\ {= \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}} \end{matrix}$ ${\left( {{\left( {C - M} \right)\quad \left( {C - M} \right)^{T}} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}} - {{\underset{\_}{p}}_{2}{\underset{\_}{p}}_{2}^{T}}} \right)\underset{\_}{p}} = {{R_{2}\underset{\_}{p}} = {{0\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}} = {\lambda_{3}{{\underset{\_}{p}}_{3}.}}}}$

This indicates that the trial eigenvector p^(T)=(0,0,1) happened to correspond to the eigenvector p₃ ^(T)=(0,0,1) belonging to the eigenvalue λ₃=0. Indeed, one may readily verify that: ${\left( {\left( {C - M} \right)\quad \left( {C - M} \right)^{T}} \right){\underset{\_}{p}}_{3}} = {{\begin{pmatrix} 2 & 1 & 0 \\ 1 & \frac{1}{2} & 0 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}} = {\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} = {{0\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}} = {\lambda_{3}{{\underset{\_}{p}}_{3}.}}}}}$

Similarly, SVD of C−M shows that C−M =PT^(T), where P is the Principal Component matrix (whose columns are orthonormalized eigenvectors proportional to p₁, p₂ and p₃, and whose elements are the Loadings, the direction cosines of the new Principal Component axes 1860, 1870 and 1880 related to the original variable axes) and T is the Scores matrix (whose rows are the coordinates of the data points 1810 and 1820, referred to the new Principal Component axes 1860, 1870 and 1880): ${C - M} = {\begin{pmatrix} \frac{2}{\sqrt{5}} & \frac{- 1}{\sqrt{5}} & 0 \\ \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} & 0 \\ 0 & 0 & 1 \end{pmatrix}\quad \begin{pmatrix} \frac{\sqrt{5}}{\sqrt{2}} & 0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix}\quad \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{- 1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}}$ ${C - M} = {{\begin{pmatrix} \frac{2}{\sqrt{5}} & \frac{- 1}{\sqrt{5}} & 0 \\ \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} & 0 \\ 0 & 0 & 1 \end{pmatrix}\quad \begin{pmatrix} \frac{\sqrt{5}}{\sqrt{2}} & \frac{- \sqrt{5}}{\sqrt{2}} \\ 0 & 0 \\ 0 & 0 \end{pmatrix}}\quad = {{PT}^{T}.}}$

The transpose of the Scores matrix (T^(T)) is given by the product of the matrix of eigenvalues of C−M with a matrix whose rows are orthonormalized eigenvectors proportional to t₁ and t₂. As shown in FIG. 18, the direction cosine (Loading) of the first Principal Component axis 1860 with respect to the variable 1 values axis is given by ${{\cos \quad \Theta_{11}} = \frac{2}{\sqrt{5}}},$

and the direction cosine (Loading) of the first Principal Component axis 1860 with respect to the variable 2 values axis is given by ${\cos \quad \Theta_{21}} = {\frac{1}{\sqrt{5}}.}$

Similarly, the direction cosine (Loading) of the first Principal Component axis 1860 with respect to the variable 3 values axis is given by ${\cos \quad \Theta_{31}} = {{\cos \quad \left( \frac{\pi}{2} \right)} = 0.}$

Similarly, the direction cosine (Loading) of the second Principal Component axis 1870 with respect to the variable 1 values axis is given by ${{\cos \quad \Theta_{12}} = \frac{- 1}{\sqrt{5}}},$

the direction cosine (Loading) of the second Principal Component axis 1870 with respect to the variable 2 values axis is given by ${{\cos \quad \Theta_{22}} = \frac{2}{\sqrt{5}}},$

and the direction cosine (Loading) of the second Principal Component axis 1870 with respect to the variable 3 values axis is given by ${\cos \quad \Theta_{32}} = {{\cos \left( \frac{\pi}{2} \right)} = 0.}$

Lastly, the direction cosine (Loading) of the third Principal Component axis 1880 with respect to the variable 1 values axis is given by ${{\cos \quad \Theta_{13}} = {{\cos \left( \frac{\pi}{2} \right)} = 0}},$

the direction cosine (Loading) of the third Principal Component axis 1880 with respect to the variable 2 values axis is given by ${{\cos \quad \Theta_{23}} = {{\cos \left( \frac{\pi}{2} \right)} = 0}},$

and the direction cosine (Loading) of the third Principal Component axis 1880 with respect to the variable 3 values axis is given by cos Θ₃₃=cos (0)=1.

SVD confirms that the singular values of C−M are 5/2 and 0, the non-negative square roots of the eigenvalues λ₁=5/2 and λ₂=0 of the matrix product (C−M)^(T)(C−M). Note that the columns of the Principal Component matrix P are the orthonormalized eigenvectors of the matrix product (C−M)(C−M)^(T).

Taking another example, a 3×4 matrix D (identical to the 3×4 matrix B^(T) given above): $D = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \end{pmatrix}$

may be taken as the overall data matrix X (again for the sake of simplicity), corresponding to 4 runs over 3 variables. As shown in FIG. 19, a scatterplot 1900 of data points with coordinates (1,1,0), (1,0,1), (1,0,−1) and (1,−1,0), respectively, may be plotted in a 3-dimensional variable space where the variables are respective rapid thermal processing tool and/or parameter values for each of the 3 variables. The mean vector 1920 μ may lie at the center of a 2-dimensional Principal Component ellipsoid 1930 (really a circle, a somewhat degenerate ellipsoid). The mean vector 1920 μ may be determined by taking the average of the columns of the overall 3×4 data matrix D. The Principal Component ellipsoid 1930 may have a first Principal Component 1940 (the “major” axis in FIG. 19, with length 2, lying along a first Principal Component axis 1950), a second Principal Component 1960 (the “minor” axis in FIG. 19, also with length 2, lying along a second Principal Component axis 1970), and no third Principal Component lying along a third Principal Component axis 1980. Here, two of the eigenvalues of the mean-scaled data matrix D−M are equal, so the lengths of the “major” and “minor” axes of the Principal Component ellipsoid 1930 in FIG. 19 are both equal, and the remaining eigenvalue is equal to zero, so the length of the other “minor” axis of the Principal Component ellipsoid 1930 in FIG. 19 is equal to zero. As shown in FIG. 19, the mean vector 1920 μ is given by: $\underset{\_}{\mu} = {{\frac{1}{4}\left\lbrack {\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \\ {- 1} \end{pmatrix} + \begin{pmatrix} 1 \\ {- 1} \\ 0 \end{pmatrix}} \right\rbrack} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}}$

and the matrix M has the mean vector 1920 μ for all 4 columns. As shown in FIG. 19, PCA is nothing more than a principal axis rotation of the original variable axes (here, the respective rapid thermal processing tool and/or parameter values for each of the 3 variables) about the endpoint of the mean vector 1920 μ, with coordinates (1,0,0) with respect to the original coordinate axes and coordinates [0,0,0] with respect to the new Principal Component axes 1950, 1970 and 1980. The Loadings are merely the direction cosines of the new Principal Component axes 1950, 1970 and 1980 with respect to the original variable axes. The Scores are simply the coordinates of the data points, [1,0,0], [0,1,0], [0,−1,0] and [−1,0,0], respectively, referred to the new Principal Component axes 1950, 1970 and 1980.

The 3×3 matrix product (D−M)(D−M)^(T) is given by: ${\left( {D - M} \right)\left( {D - M} \right)^{T}} = {{\begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \end{pmatrix}\quad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & {- 1} \\ 0 & {- 1} & 0 \end{pmatrix}} = {\begin{pmatrix} 0 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}.}}$

The 3×3 matrix (D−M)(D−M)^(T) is 3 times the covariance matrix S_(3×3), having elements s_(ij), where i=1,2,3, and j=1,2,3, defined so that: ${s_{ij} = \frac{{4{\sum\limits_{k = 1}^{4}{d_{ik}d_{jk}}}} - {\sum\limits_{k = 1}^{4}{d_{ik}{\sum\limits_{k = 1}^{4}d_{jk}}}}}{4\left( {4 - 1} \right)}},$

corresponding to the rectangular 3×4 matrix D_(3×4).

EIG reveals that the eigenvalues of the matrix product (D−M)(D−M)^(T) are 0, 2 and 2. The eigenvectors of the matrix product (D−M)(D−M)^(T) are solutions p of the equation ((D−M)(D−M)^(T))p=λp, and may be seen by inspection to be p₁ ^(T)=(0,1,0), p₂ ^(T)=(0,0,1), and p₃ ^(T)=(1,0,0), belonging to the eigenvalues λ₁=2, λ₂=2, and λ₃=0, respectively (following the convention of placing the largest eigenvalue first).

As may be seen in FIG. 19, the direction cosine (Loading) of the first Principal Component axis 1950 with respect to the variable 1 values axis is given by ${{\cos \quad \Theta_{11}} = {{\cos \left( \frac{\pi}{2} \right)} = 0}},$

the direction cosine (Loading) of the first Principal Component axis 1970 with respect to the variable 2 values axis is given by cosΘ₂₁=cos (0)=1, and the direction cosine (Loading) of the first Principal Component axis 1060 with respect to the variable 3 values axis is given by ${\cos \quad \Theta_{31}} = {{\cos \left( \frac{\pi}{2} \right)} = 0.}$

Similarly, the direction cosine (Loading) of the second Principal Component axis 1970 with respect to the variable 1 values axis is given by ${{\cos \quad \Theta_{12}} = {{\cos \quad \left( \frac{\pi}{2} \right)} = 0}},$

the direction cosine (Loading) of the second Principal Component axis 1970 with respect to the variable 2 values axis is given by ${{\cos \quad \Theta_{22}} = {{\cos \quad \left( \frac{\pi}{2} \right)} = 0}},$

and the direction cosine (Loading) of the second Principal Component axis 1970 with respect to the variable 3 values axis is given by cos Θ₃₂=cos (0)=1. Lastly, the direction cosine (Loading) of the third Principal Component axis 1980 with respect to the variable 1 values axis is given by cosΘ₁₃=cos (0)=1, the direction cosine (Loading) of the third Principal Component axis 1980 with respect to the variable 2 values axis is given by ${{\cos \quad \Theta_{23}} = {{\cos \quad \left( \frac{\pi}{2} \right)} = 0}},$

and the direction cosine (Loading) of the third Principal Component axis 1980 with respect to the variable 3 values axis is given by ${\cos \quad \Theta_{33}} = {{\cos \quad \left( \frac{\pi}{2} \right)} = 0.}$

The transpose of the Scores matrix T^(T) may be obtained simply by multiplying the mean-scaled data matrix D−M on the left by the transpose of the Principal Component matrix P, whose columns are p₁, p₂, p₃, the orthonormalized eigenvectors of the matrix product (D−M)(D−M)^(T); $\begin{matrix} {T^{T} = {{P^{T}\left( {D - M} \right)} = {\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\quad \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \end{pmatrix}}}} \\ {= {\begin{pmatrix} 1 & 0 & 0 & {- 1} \\ 0 & 1 & {- 1} & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}.}} \end{matrix}$

The columns of the transpose of the Scores matrix T^(T) (or, equivalently, the rows of the Scores matrix T) are, indeed, the coordinates of the data points, [1,0,0], [0,1,0], [0,−1,0] and [−1,0,0], respectively, referred to the new Principal Component axes 1950, 1970 and 1980.

The matrices C and D discussed above have been used for the sake of simplifying the presentation of PCA and the power method, and are much smaller than the data matrices encountered in illustrative embodiments of the present invention. For example, in various illustrative embodiments, about m=20-100 rapid thermal processing runs may be measured and/or monitored over about n=4-100 rapid thermal processing tool variables and/or rapid thermal processing parameters. Brute force modeling, regressing all m=20-100 runs over n=4-100 variables, may constitute an ill-conditioned regression problem. Techniques such as PCA and/or partial least squares (PLS, also known as projection to latent structures) reduce the complexity in such cases by revealing the hierarchical ordering of the data based on levels of decreasing variability. In PCA, this involves finding successive Principal Components. In PLS techniques such as NIPALS, this involves finding successive latent vectors. In various illustrative embodiments, the tool and/or sensor drift during about m=20-100 rapid thermal processing runs measured and/or monitored over about n=4−100 rapid thermal processing tool variables and/or rapid thermal processing parameters may be mapped to an equivalent problem of the dynamic flow of about m=20-100 points (representing the about m=20-100 rapid thermal processing runs) through an n-dimensional space (representing the about n=4-100 variables). PCA may be used, for example, to correct the rapid thermal processing by indicating an appropriate multi-dimensional “rotation” to be made on the rapid thermal processing tool variables and/or rapid thermal processing parameters to compensate for the tool and/or sensor drift from the respective setpoint values.

In various alternative illustrative embodiments, first-principles radiation models may be built in alternative ways. Such first-principles radiation models may also be formed by monitoring one or more tool variables and/or one or more rapid thermal processing parameters during one or more rapid thermal processing runs. Examples of such tool variables and/or rapid thermal processing parameters may comprise one or more pyrometer trace readings, one or more lamp power trace readings, one or more tube temperature trace readings, gas composition, gas flow rates, pressure, and the like. In these various alternative illustrative embodiments, building the first-principles radiation models may comprise fitting the collected processing data using at least one of polynomial curve fitting, least-squares fitting, polynomial least-squares fitting, non-polynomial least-squares fitting, weighted least-squares fitting, weighted polynomial least-squares fitting, and weighted non-polynomial least-squares fitting, either in addition to, or as an alternative to, using Partial Least Squares (PLS) and/or Principal Components Analysis (PCA), as described above.

In various illustrative embodiments, samples may be collected for N+1 data points (x_(i),y_(i)), where i=1, 2, . . . , N, N+1, and a polynomial of degree N, ${{P_{N}(x)} = {{a_{0} + {a_{1}x} + {a_{2}x^{2}} + \ldots \quad + {a_{k}x^{k}} + \ldots \quad + {a_{N}x^{N}}} = {\sum\limits_{k = 0}^{N}{a_{k}x^{k}}}}},$

may be fit to the N+1 data points (x_(i),y_(i)). For example, 100 time data points (N=99) may be taken relating the pyrometer trace reading p, the lamp power trace reading f, and/or the tube temperature trace reading T, during a rapid thermal processing step, to the effective yield t of workpieces emerging from the rapid thermal processing step, resulting in respective sets of N+1 data points (p_(i),t_(i)), (f_(i),t_(i)), and/or (T_(i),t_(i)). The values may be the actually measured values of the rapid thermal processing tool variables and/or rapid thermal processing parameters, or ratios of actually measured values (normalized to respective reference setpoints), or logarithms of such ratios, for example. Polynomial interpolation is described, for example, in Numerical Methods for Scientists and Engineers, by R. W. Hamming, Dover Publications, New York, 1986, at pages 230-235. The requirement that the polynomial P_(N)(x) pass through the N+1 data points (x_(i),y_(i)) is ${y_{i} = {{P_{N}\left( x_{i} \right)} = {\sum\limits_{k = 0}^{N}{a_{k}x_{i}^{k}}}}},{{{for}\quad i} = 1},2,\ldots \quad,N,{N + 1},$

a set of N+1 conditions. These N+1 conditions then completely determine the N+1 coefficients a_(k), for k=0, 1, . . . , N.

The determinant of the coefficients of the unknown coefficients a_(k) is the Vandermonde determinant: ${V_{N + 1} = {{\begin{matrix} 1 & x_{1} & x_{1}^{2} & \cdots & x_{1}^{N} \\ 1 & x_{2} & x_{2}^{2} & \cdots & x_{2}^{N} \\ \vdots & \vdots & \vdots & ⋰ & \vdots \\ 1 & x_{N} & x_{N}^{2} & \cdots & x_{N}^{N} \\ 1 & x_{N + 1} & x_{N + 1}^{2} & \cdots & x_{N + 1}^{N} \end{matrix}} = {x_{i}^{k}}}},$

where i=1, 2, . . . , N+1, and k=0, 1, . . . , N. The Vandermonde determinant V_(N+1), considered as a function of the variables x_(i), V_(N+1)=V_(N+1)(x₁,x₂, . . . ,x_(N),x_(N+1)), is clearly a polynomial in the variables x_(i), as may be seen by expanding out the determinant, and a count of the exponents shows that the degree of the polynomial is ${0 + 1 + 2 + 3 + \ldots \quad + k + \ldots + N} = {{\sum\limits_{k = 0}^{N}k} = \frac{N\left( {N + 1} \right)}{2}}$

(for example, the diagonal term of the Vandermonde determinant V_(N+1) is 1·x₂·x₃ ² . . . x_(N) ^(N−1)·x_(N+1) ^(N)).

Now, if x_(N+1)=x_(j), for j=1, 2, . . . , N, then the Vandermonde determinant V_(N+1)=0, since any determinant with two identical rows vanishes, so the Vandermonde determinant V_(N+1) must have the factors (x_(N+1)−x_(j)), for j=1, 2, . . . , N, corresponding to the N factors $\prod\limits_{j = 1}^{N}{\left( {x_{N + 1} - x_{j}} \right).}$

Similarly, if x_(N)=x_(j), for j=1, 2, . . . , N−1, then the Vandermonde determinant V_(N+1)=0. So the Vandermonde determinant V_(N+1) must also have the factors (x_(N)−x_(j)), for j=1, 2, . . . , N−1, corresponding to the N−1 factors $\prod\limits_{j = 1}^{N - 1}\quad {\left( {x_{N} - x_{j}} \right).}$

Generally, if x_(m)=x_(j), for j<m, where m=2, . . . , N, N+1, then the Vandermonde determinant V_(N+1)=0, so the Vandermonde determinant V_(N+1) must have all the factors (x_(m)−x_(j)), for j<m, where m=2, . . . , N, N+1, corresponding to the factors $\prod\limits_{{m > j} = 1}^{N + 1}\quad {\left( {x_{m} - x_{j}} \right).}$

Altogether, this represents a polynomial of degree ${{N + \left( {N - 1} \right) + \quad \ldots \quad + k + \quad \ldots \quad + 2 + 1} = {{\sum\limits_{k = 1}^{N}k} = \frac{N\left( {N + 1} \right)}{2}}},$

since, when m=N+1, for example, j may take on any of N values, j=1, 2, . . . , N, and when m=N, j may take on any of N−1 values, j=1, 2, . . . , N−1, and so forth (for example, when m=3, j may take only two values, j=1, 2, and when m=2, j may take only one value, j=1), which means that all the factors have been accounted for and all that remains is to find any multiplicative constant by which these two representations for the Vandermonde determinant V_(N+1) might differ. As noted above, the diagonal term of the Vandermonde determinant V_(N+1) is 1·x₂·x₃ ² . . . x_(N) ^(N−1)·x_(N+1) ^(N), and this may be compared to the term from the left-hand sides of the product of factors ${{\prod\limits_{{m > j} = 1}^{N + 1}\quad \left( {x_{m} - x_{j}} \right)} = {\prod\limits_{j = 1}^{N}{\left( {x_{N + 1} - x_{j}} \right){\prod\limits_{j = 1}^{N - 1}{\left( {x_{N} - x_{j}} \right)\quad \ldots \quad {\prod\limits_{j = 1}^{2}{\left( {x_{3} - x_{J}} \right){\prod\limits_{j = 1}^{I}\left( {x_{2} - x_{J}} \right)}}}}}}}},\quad {{x_{N + 1}^{N} \cdot x_{N}^{N - 1}}\quad \ldots \quad {x_{3}^{2} \cdot x_{2}}},$

which is identical, so the multiplicative constant is unity and the Vandermonde determinant ${V_{N + 1}\quad {is}\quad {V_{N + 1}\left( {x_{1},x_{2},\ldots \quad,x_{N + 1}} \right)}} = {{x_{i}^{k}} = {\prod\limits_{{m > j} = 1}^{N + 1}{\left( {x_{m} - x_{j}} \right).}}}$

This factorization of the Vandermonde determinant V_(N+1) shows that if x_(i)≠x_(j), for i≠j, then the Vandermonde determinant V_(N+1), cannot be zero, which means that it is always possible to solve for the unknown coefficients a_(k), since the Vandermonde determinant V_(N+1) is the determinant of the coefficients of the unknown coefficients a_(k). Solving for the unknown coefficients a_(k), using determinants, for example, substituting the results into the polynomial of degree N, ${{P_{N}(x)} = {\sum\limits_{k = 0}^{N}{a_{k}x^{k}}}},$

and rearranging suitably gives the determinant equation ${{\begin{matrix} y & 1 & x & x^{2} & \ldots & x^{N} \\ y_{1} & 1 & x_{1} & x_{1}^{2} & \ldots & x_{1}^{N} \\ y_{2} & 1 & x_{2} & x_{2}^{2} & \ldots & x_{2}^{N} \\ \vdots & \vdots & \vdots & \vdots & ⋰ & \vdots \\ y_{N} & 1 & x_{N} & x_{N}^{2} & \ldots & x_{N}^{N} \\ y_{N + 1} & 1 & x_{N + 1} & x_{N + 1}^{2} & \ldots & x_{N + 1}^{N} \end{matrix}} = 0},$

which is the solution to the polynomial fit. This may be seen directly as follows. Expanding this determinant by the elements of the top row, this is clearly a polynomial of degree N. The coefficient of the element y in the first row in the expansion of this determinant by the elements of the top row is none other than the Vandermonde determinant V_(N+1). In other words, the cofactor of the element y in the first row is, in fact, the Vandermonde determinant V_(N+1). Indeed, the cofactor of the nth element in the first row, where n=2, . . . , N+2, is the product of the coefficient a_(n−2) in the polynomial expansion $y = {{P_{N}(x)} = {\sum\limits_{k = 0}^{N}{a_{k}x^{k}}}}$

with the Vandermonde determinant V_(N+1). Furthermore, if x and y take on any of the sample values x_(i) and y_(i), for i=1, 2, . . . , N, N+1, then two rows of the determinant would be the same and the determinant must then vanish. Thus, the requirement that the polynomial y=P_(N)(x) pass through the N+1 data points (x_(i),y_(i)), ${y_{i} = {{P_{N}\left( x_{i} \right)} = {\sum\limits_{k = 0}^{N}{a_{k}x_{i}^{k}}}}},\quad {{{for}\quad i} = 1},2,\ldots \quad,N,{N + 1},$

is satisfied.

For example, a quadratic curve may be found that goes through the sample data set (−1,a), (0,b), and (1,c). The three equations are P₂(−1)=a=a₀−a₁+a₂, P₂ (0)=b=a₀, and P₂(1)=c=a₀+a₁+a₂, which imply that b=a₀, c−a=2a₁, and c+a−2b=2a₂, so that ${{y(x)} = {{P_{2}(x)} = {b + {\frac{c - a}{2}x} + {\frac{c + a - {2b}}{2}x^{2}}}}},$

which is also the result of expanding $\begin{matrix} {{\begin{matrix} y & 1 & x & x^{2} \\ a & 1 & {- 1} & 1 \\ b & 1 & 0 & 0 \\ c & 1 & 1 & 1 \end{matrix}} = {0 = {{y{\begin{matrix} 1 & {- 1} & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \end{matrix}}} - {1{\begin{matrix} a & {- 1} & 1 \\ b & 0 & 0 \\ c & 1 & 1 \end{matrix}}} +}}} \\ {{{{x{\begin{matrix} a & 1 & 1 \\ b & 1 & 0 \\ c & 1 & 1 \end{matrix}}} - {x^{2}{\begin{matrix} a & 1 & {- 1} \\ b & 1 & 0 \\ c & 1 & 1 \end{matrix}}}},}} \end{matrix}$

the coefficient of y being the respective Vandermonde determinant V₃=2.

Similarly, a quartic curve may be found that goes through the sample data set (−2,a), (−1,b), (0,c), (1,b), and (2,a). The five equations are P₄(−2)=a=a₀−2a₁+4a₂−8a₃+16a₄, P₄(−1)=b=a₀−a₁+a₂−a₃+a₄, P₄(0)=c=a₀, P₄(1)=b=a₀+a₁+a₂+a₃+a₄, and P₄(2)=a=a₀+2a₁+4a₂+8a₃+16a₄, which imply that c=a₀, 0=a₁=a₃ (which also follows from the symmetry of the data set), (a−c)−16(b−c)=−12a₂, and (a−c)−4(b−c)=12a₄, so that ${y(x)} = {{P_{4}(x)} = {c - {\frac{a - {16b} + {15c}}{12}\quad x^{2}} + {\frac{a - {4b} + {3c}}{12}{x^{4}.}}}}$

In various alternative illustrative embodiments, samples may be collected for M data points (x_(i),y_(i)), where i=1, 2, . . . , M, and a first degree polynomial (a straight line), ${{P_{1}(x)} = {{a_{0} + {a_{1}x}} = {\sum\limits_{k = 0}^{1}{a_{k}x^{k}}}}},$

may be fit (in a least-squares sense) to the M data points (x_(i),y_(i)). For example, 100 time data points (M=100) may be taken relating the pyrometer trace reading p, the lamp power trace reading ƒ, and/or the tube temperature trace reading T, during a rapid thermal processing step, to the effective yield t of workpieces emerging from the rapid thermal processing step, resulting in the M data points (p_(i),t_(i)), (ƒ_(i),t_(i)), and/or (T_(i),t_(i)). The values may be the actually measured values of the rapid thermal processing tool variables and/or rapid thermal processing parameters, or ratios of actually measured values (normalized to respective reference setpoints), or logarithms of such ratios, for example. Least-squares fitting is described, for example, in Numerical Methods for Scientists and Engineers, by R. W. Hamming, Dover Publications, New York, 1986, at pages 427-443.

The least-squares criterion may be used in situations where there is much more data available than parameters so that exact matching (to within round-off) is out of the question. Polynomials are most commonly used in least-squares matching, although any linear family of suitable functions may work as well. Suppose some quantity x is being measured by making M measurements x_(i), for i=1, 2, . . . , M, and suppose that the measurements x_(i), are related to the “true” quantity x by the relation x_(i)=x+ε_(i), for i=1, 2, . . . , M, where the residuals ε_(i) are regarded as noise. The principle of least-squares states that the best estimate ξ of the true value x is the number that minimizes the sum of the squares of the deviations of the data from their estimate ${{f(\xi)} = {{\sum\limits_{i = 1}^{M}ɛ_{i}^{2}} = {\sum\limits_{i = 1}^{M}\left( {x_{i} - \xi} \right)^{2}}}},$

which is equivalent to the assumption that the average x_(a), where ${x_{a} = {\frac{1}{M}{\sum\limits_{i = 1}^{M}x_{i}}}},$

is the best estimate ξ of the true value x. This equivalence may be shown as follows. First, the principle of least-squares leads to the average x_(a). Regarding ${f(\xi)} = {{\sum\limits_{i = 1}^{M}ɛ_{i}^{2}} = {\sum\limits_{i = 1}^{M}\left( {x_{i} - \xi} \right)^{2}}}$

as a function of the best estimate ξ, minimization with respect to the best estimate ξ may proceed by differentiation: ${\frac{{f(\xi)}}{\xi} = {{{- 2}{\sum\limits_{i = 1}^{M}\left( {x_{i} - \xi} \right)}} = 0}},$

which implies that ${{{\sum\limits_{i = 1}^{M}x_{i}} - {\sum\limits_{i = 1}^{M}\xi}} = {0 = {{\sum\limits_{i = 1}^{M}x_{i}} - {M\quad \xi}}}},$

so that ${\xi = {{\frac{1}{M}{\sum\limits_{i = 1}^{M}x_{i}}} = x_{a}}},$

or, in other words, that the choice x_(a)=ξ minimizes the sum of the squares of the residuals ε_(i). Noting also that ${\frac{d^{2}{f(\xi)}}{d\quad \xi^{2}} = {{2{\sum\limits_{i = 1}^{M}1}} = {{2M} > 0}}},$

the criterion for a minimum is established.

Conversely, if the average x_(a) is picked as the best choice x_(a)=ξ, it can be shown that this choice, indeed, minimizes the sum of the squares of the residuals ε_(i). Set $\begin{matrix} {{f\left( x_{a} \right)} = {{\prod\limits_{i = 1}^{M}\left( {x_{i} - x_{a}} \right)^{2}} = {{\sum\limits_{i = 1}^{M}x_{i}^{2}} - {2x_{a}{\sum\limits_{i = 1}^{M}x_{i}}} + {\sum\limits_{i = 1}^{M}x_{a}^{2}}}}} \\ {= {{\sum\limits_{i = 1}^{M}x_{i}^{2}} - {2x_{a}M\quad x_{a}} + {M\quad x_{a}^{2}}}} \\ {= {{\sum\limits_{i = 1}^{M}x_{i}^{2}} - {M\quad {x_{a}^{2}.}}}} \end{matrix}$

If any other value x_(b) is picked, then, plugging that other value x_(b) into ƒ(x) gives $\begin{matrix} {{f\left( x_{b} \right)} = {{\sum\limits_{i = 1}^{M}\left( {x_{i} - x_{b}} \right)^{2}} = {{\sum\limits_{i = 1}^{M}x_{i}^{2}} - {2x_{b}{\sum\limits_{i = 1}^{M}x_{i}}} + {\sum\limits_{i = 1}^{M}x_{b}^{2}}}}} \\ {= {{\sum\limits_{i = 1}^{M}x_{i}^{2}} - {2x_{b}M\quad x_{a}} + {M\quad {x_{b}^{2}.}}}} \end{matrix}$

Subtracting ƒ(x_(a)) from ƒ(x_(b)) gives ƒ(x_(b))−ƒ(x_(a))=M[x_(a) ²−2x_(a)x_(b)+x_(b) ²]=M(x_(a)−x_(b))²≧0, so that ƒ(x_(b))≧ƒ(x_(a)), with equality if, and only if, x_(b)=x_(a). In other words, the average x_(a), indeed, minimizes the sum of the squares of the residuals ε_(i). Thus, it has been shown that the principle of least-squares and the choice of the average as the best estimate are equivalent.

There may be other choices besides the least-squares choice. Again, suppose some quantity x is being measured by making M measurements x_(i), for i=1, 2, . . . , M, and suppose that the measurements x_(i), are related to the “true” quantity x by the relation x_(i)=x+ε_(i), for i=1, 2, . . . , M, where the residuals ε_(i) are regarded as noise. An alternative to the least-squares choice may be that another estimate χ of the true value x is the number that minimizes the sum of the absolute values of the deviations of the data from their estimate ${{f(\chi)} = {{\sum\limits_{i = 1}^{M}{ɛ_{i}}} = {\sum\limits_{i = 1}^{M}{{x_{i} - \chi}}}}},$

which is equivalent to the assumption that the median or middle value x_(m) of the M measurements x_(i), for i=1, 2, . . . , M (if M is even, then average the two middle values), is the other estimate χ of the true value x. Suppose that there are an odd number M=2k+1 of measurements x_(i), for i=1, 2, . . . , M, and choose the median or middle value x_(m) as the estimate χ of the true value x that minimizes the sum of the absolute values of the residuals ε_(i). Any upward shift in this value x_(m) would increase the k terms |x_(i)−x| that have x_(i) below x_(m), and would decrease the k terms |x_(i)−x| that have x_(i) above x_(m), each by the same amount. However, the upward shift in this value x_(m) would also increase the term |x_(m)−x| and, thus, increase the sum of the absolute values of all the residuals ε_(i). Yet another choice, instead of minimizing the sum of the squares of the residuals ε_(i), would be to choose to minimize the maximum deviation, which leads to ${\frac{x_{\min} + x_{\max}}{2} = x_{midrange}},$

the midrange estimate of the best value.

Returning to the various alternative illustrative embodiments in which samples may be collected for M data points (x_(i),y_(i)), where i=1, 2, . . . , M, and a first degree polynomial (a straight line), ${{P_{1}(x)} = {{a_{0} + {a_{1}x}} = {\sum\limits_{k = 0}^{1}{a_{k}x^{k}}}}},$

may be fit (in a least-squares sense) to the M data points (x_(i),y_(i)), there are two parameters, a₀ and a₁, and a function F(a₀,a₁) that needs to be minimized as follows. The function F(a₀,a₁) is given by ${F\left( {a_{0},a_{1}} \right)} = {{\sum\limits_{i = 1}^{M}ɛ_{i}^{2}} = {{\sum\limits_{i = 1}^{M}\left\lbrack {{P_{1}\left( x_{i} \right)} - y_{i}} \right\rbrack^{2}} = {\sum\limits_{i = 1}^{M}\left\lbrack {a_{0} + {a_{1}x_{i}} - y_{i}} \right\rbrack^{2}}}}$

and setting the partial derivatives of F(a₀,a₁) with respect to a₀ and a₁ equal to zero gives $\frac{\partial{F\left( {a_{0},a_{1}} \right)}}{\partial a_{0}} = {{2{\sum\limits_{i = 1}^{M}\left\lbrack {a_{0} + {a_{1}x_{i}} - y_{i}} \right\rbrack}} = 0}$

and ${\frac{\partial{F\left( {a_{0},a_{1}} \right)}}{\partial a_{1}} = {{2{\sum\limits_{i = 1}^{M}{\left\lbrack {a_{0} + {a_{1}x_{i}} - y_{i}} \right\rbrack x_{i}}}} = 0}},$

respectively. Simplifying and rearranging gives ${{{a_{0}M} + {a_{1}{\sum\limits_{i = 1}^{M}x_{i}}}} = {\sum\limits_{i = 1}^{M}{y_{i}\quad {and}}}}\quad$ ${{{a_{0}{\sum\limits_{i = 1}^{M}x_{i}}} + {a_{1}{\sum\limits_{i = 1}^{M}x_{i}^{2}}}} = {\sum\limits_{i = 1}^{M}{x_{i}y_{i}}}},$

respectively, where there are two equations for the two unknown parameters a₀ and a₁, readily yielding a solution.

As shown in FIG. 20, for example, a first degree polynomial (a straight line), ${{P_{1}(x)} = {{a_{0} + {a_{1}x}} = {\sum\limits_{k = 0}^{1}{a_{k}x^{k}}}}},$

may be fit (in a least-squares sense) to the M=5 data points (1,0), (2,2), (3,2), (4,5), and (5,4). The residuals ε_(i), for i=1, 2, . . . , 5, are schematically illustrated in FIG. 20. The equations for the two parameters a₀ and a₁ are 5a₀+15a₁=13 and 15a₀+55a₁=50, respectively, so that, upon multiplying the first equation by 3 and then subtracting that from the second equation, thereby eliminating a₀, the solution for the parameter a₁ becomes a₁=11/10, and this implies that the solution for the parameter a₀ becomes a₀=−7/10. The first degree polynomial (the straight line) that provides the best fit, in the least-squares sense, is ${{P_{1}(x)} = {{{- \quad \frac{7}{10}} + {\frac{11}{10}x}} = {\frac{1}{10}\left( {{- 7} + {11x}} \right)}}},$

as shown in FIG. 20.

As shown in FIG. 21, for example, a first degree polynomial (a straight line), ${{P_{1}(x)} = {{a_{0} + {a_{1}x}} = {\sum\limits_{k = 0}^{1}{a_{k}x^{k}}}}},$

may be fit (in a least-squares sense) to the M=7 data points (−3,4), (−2,4), (−1,2), (0,2), (1,1), (2,0), and (3,0). The residuals ε_(i), for i=1, 2, . . . , 7, are schematically illustrated in FIG. 21. The equations for the two parameters a₀ and a₁ are ${{a_{0}M} + {a_{1}{\sum\limits_{i = 1}^{M}x_{i}}}} = {{\sum\limits_{i = 1}^{M}y_{i}} = {{{7a_{0}} + {a_{1}\left( {{- 3} - 2 - 1 + 0 + 1 + 2 + 3} \right)}} = \left( {4 + 4 + 2 + 2 + 1 + 0 + 0} \right)}}$ and ${{{a_{0}{\sum\limits_{i = 1}^{M}x_{i}}} + {a_{1}{\sum\limits_{i = 1}^{M}x_{i}^{2}}}} = {{\sum\limits_{i = 1}^{M}{x_{i}y_{i}}} = {{a_{1}\left( {9 + 4 + 1 + 0 + 1 + 4 + 9} \right)} = \left( {{- 12} - 8 - 2 + 0 + 1 + 0 + 0} \right)}}},$

respectively, which give 7a₀=13 and 28a₁=−21, respectively. In other words, a₀=13/7 and a₁=−3/4, so that, the first degree polynomial (the straight line) that provides the best fit, in the least-squares sense, is ${{P_{1}(x)} = {\frac{13}{7} - {\frac{3}{4}x}}},$

as shown in FIG. 21.

In various other alternative illustrative embodiments, samples may be collected for M data points (x_(i),y_(i)), where i=1, 2, . . . , M, and a polynomial of degree N, ${{P_{N}(x)} = {{a_{0} + {a_{1}x} + {a_{2}x^{2}} + \ldots + {a_{k}x^{k}} + \ldots + {a_{N}x^{N}}} = {\sum\limits_{k = 0}^{N}{a_{k}x^{k}}}}},$

may be fit (in a least-squares sense) to the M data points (x_(i),y_(i)). For example, 100 time data points (M=100) may be taken relating the pyrometer trace reading p, the lamp power trace reading ƒ, and/or the tube temperature trace reading T, during a rapid thermal processing step, to the effective yield t of workpieces emerging from the rapid thermal processing step, resulting in the M data points (p_(i),t_(i)), (ƒ_(i),t_(i)), and/or (T_(i),t_(i)). The values may be the actually measured values of the rapid thermal processing tool variables and/or rapid thermal processing parameters, or ratios of actually measured values (normalized to respective reference setpoints), or logarithms of such ratios, for example. In one illustrative embodiment, the degree N of the polynomial is at least 10 times smaller than M. Representative values for the number M of data points lie in a range of about 10 to about 1000, and representative values for the degree N of the polynomials is in a range of about 1 to about 10.

The function F(a₀,a₁, . . . ,a_(N)) may be minimized as follows. The function F(a₀,a₁, . . . ,a_(N)) is given by ${F\left( {a_{0},a_{1},\ldots \quad,a_{N}} \right)} = {{\sum\limits_{i = 1}^{M}ɛ_{i}^{2}} = {\sum\limits_{i = 1}^{M}\left\lbrack {{P_{N}\left( x_{i} \right)} - y_{i}} \right\rbrack^{2}}}$

and setting the partial derivatives of F(a₀,a₁, . . . ,a_(N)) with respect to a_(j), for j=0, 1, . . . , N, equal to zero gives ${\frac{\partial{F\left( {a_{0},a_{1},\ldots \quad,a_{N}} \right)}}{\partial a_{j}} = {{2{\sum\limits_{i = 1}^{M}{\left\lbrack {{P_{N}\left( x_{i} \right)} - y_{i}} \right\rbrack x_{i}^{j}}}} = {{2{\sum\limits_{i = 1}^{M}{\left\lbrack {{\sum\limits_{k = 0}^{N}{a_{k}x_{i}^{k}}} - y_{i}} \right\rbrack x_{i}^{j}}}} = 0}}},$

for j=0, 1, . . . , N, since (x_(i))^(j) is the coefficient of a_(j) in the polynomial ${P_{N}\left( x_{i} \right)} = {\sum\limits_{k = 0}^{N}{a_{k}{x_{j}^{k}.}}}$

Simplifying and rearranging gives ${{\sum\limits_{i = 1}^{M}{\left\lbrack {\sum\limits_{k = 0}^{N}{a_{k}x_{i}^{k}}} \right\rbrack x_{i}^{j}}} = {{{\sum\limits_{k = 0}^{N}{a_{k}\left\lbrack {\sum\limits_{i = 1}^{M}x_{i}^{k + j}} \right\rbrack}} \equiv {\sum\limits_{k = 0}^{N}{a_{k}S_{k + j}}}} = {{\sum\limits_{i = 1}^{M}{x_{i}^{j}y_{i}}} \equiv T_{j}}}},$

for j=0, 1, . . . , N, where ${{\sum\limits_{i = 1}^{M}x_{i}^{k + j}} \equiv {S_{k + j}\quad \text{and}\quad {\sum\limits_{i = 1}^{M}{x_{i}^{j}y_{i}}}} \equiv T_{j}},$

respectively. There are N+1 equations ${{\sum\limits_{k = 0}^{N}{a_{k}S_{k + j}}} = T_{j}},$

for j=0, 1, . . . , N, also known as the normal equations, for the N+1 unknown parameters a_(k), for k=0, 1, . . . , N, readily yielding a solution, provided that the determinant of the normal equations is not zero. This may be demonstrated by showing that the homogeneous equations ${\sum\limits_{k = 0}^{N}{a_{k}S_{k + j}}} = 0$

only have the trivial solution a_(k)=0, for k=0, 1, . . . N, which may be shown as follows. Multiply the jth homogeneous equation by a_(j) and sum over all j, from j=0 to j=N, ${{\sum\limits_{j = 0}^{N}{a_{j}{\sum\limits_{k = 0}^{N}{a_{k}S_{k + j}}}}} = {{\sum\limits_{j = 0}^{N}{a_{j}{\sum\limits_{k = 0}^{N}{a_{k}{\sum\limits_{i = 1}^{M}{x_{i}^{k}x_{i}^{j}}}}}}} = {{\sum\limits_{i = 1}^{M}{\left( {\sum\limits_{k = 0}^{N}{a_{k}x_{i}^{k}}} \right)\left( {\sum\limits_{j = 0}^{N}{a_{j}x_{i}^{j}}} \right)}} = {{\sum\limits_{i = 1}^{M}\left( {P_{N}\left( x_{i} \right)} \right)^{2}} = 0}}}},$

which would imply that P_(N)(x_(i))≡0, and, hence, that a_(k)=0, for k=0, 1, . . . , N, the trivial solution. Therefore, the determinant of the normal equations is not zero, and the normal equations may be solved for the N+1 parameters a_(k), for k=0, 1, . . . , N, the coefficients of the least-squares polynomial of degree N, ${{P_{N}(x)} = {\sum\limits_{k = 0}^{N}{a_{k}x^{k}}}},$

that may be fit to the M data points (x_(i),y_(i)).

Finding the least-squares polynomial of degree N, ${{P_{N}(x)} = {\sum\limits_{k = 0}^{N}{a_{k}x^{k}}}},$

that may be fit to the M data points (x_(i),y_(i)) may not be easy when the degree N of the least-squares polynomial is very large. The N+1 normal equations ${{\sum\limits_{k = 0}^{N}{a_{k}S_{k + j}}} = T_{j}},$

for the j=0, 1, . . . , N+1 unknown parameters a_(k), for k=0, 1, . . . , N, may not be easy to solve, for example, when the degree N of the least-squares polynomial is much greater than about 10. This may be demonstrated as follows. Suppose that the M data points (x_(i), y_(i)) are more or less uniformly distributed in the interval 0≦x≦1, so that $S_{k + j} = {{{\sum\limits_{i = 0}^{M}x_{i}^{k + j}} \approx {M{\int_{0}^{1}{x^{k + j}{x}}}}} = {\frac{M}{k + j + 1}.}}$

The resulting determinant for the normal equations is then approximately given by ${{{S_{k + j}} \approx {\frac{M}{k + j + 1}} \approx {M^{N + 1}{\frac{1}{k + j + 1}}}} = {M^{N + 1}H_{N + 1}}},$

for j,k=0, 1, . . . , N, where H_(N), for j,k=0, 1, . . . , N−1, is the Hilbert determinant of order N, which has the value $H_{N} = \frac{\left\lbrack {{0!}{1!}{2!}{3!}\quad \ldots \quad {\left( {N - 1} \right)!}} \right\rbrack^{3}}{{N!}{\left( {N + 1} \right)!}{\left( {N + 2} \right)!}\quad \ldots \quad {\left( {{2N} - 1} \right)!}}$

that approaches zero very rapidly. For example, ${H_{1} = {\frac{\left\lbrack {0!} \right\rbrack^{3}}{1!} = {{1} = 1}}},\quad {H_{2} = {\frac{\left\lbrack {{0!}{1!}} \right\rbrack^{3}}{{2!}{3!}} = {{\begin{matrix} 1 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{3} \end{matrix}} = {{\frac{1}{3} - \frac{1}{4}} = \frac{1}{12}}}}},{{\text{and}\quad H_{3}} = {\frac{\left\lbrack {{0!}{1!}{2!}} \right\rbrack^{3}}{{3!}{4!}{5!}} = {\begin{matrix} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \end{matrix}}}},$

where ${\begin{matrix} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \end{matrix}} = {{{\frac{1}{3}{\begin{matrix} \frac{1}{2} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{4} \end{matrix}}} - {\frac{1}{4}{\begin{matrix} 1 & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{4} \end{matrix}}} + {\frac{1}{5}{\begin{matrix} 1 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{3} \end{matrix}}}} = {{{\frac{1}{3}\frac{1}{72}} - {\frac{1}{4}\frac{1}{12}} + {\frac{1}{5}\frac{1}{12}}} = {{\frac{1}{216} - \frac{1}{240}} = {\frac{1}{2160}.}}}}$

This suggests that the system of normal equations is ill-conditioned and, hence, difficult to solve when the degree N of the least-squares polynomial is very large. Sets of orthogonal polynomials tend to be better behaved.

As shown in FIG. 22, for example, a second degree polynomial (a quadratic), ${{P_{2}(x)} = {{a_{0} + {a_{1}x} + {a_{2}x^{2}}} = {\sum\limits_{k = 0}^{2}{a_{k}x^{k}}}}},$

may be fit (in a least-squares sense) to the M=7 data points (−3,4), (−2,2), (−1,3), (0,0), (1,−1), (2,−2), and (3,−5). The residuals ε_(i), for i=1, 2, . . . , 7, are schematically illustrated in FIG. 22. The three normal equations are ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k + j}}} = T_{j}},$

for j =0, 1, 2, where ${{\sum\limits_{i = 1}^{7}x_{i}^{k + j}} \equiv {S_{k + j}\quad \text{and}\quad {\sum\limits_{i = 1}^{7}{x_{i}^{j}y_{i}}}} \equiv T_{j}},$

respectively, for the three parameters a₀, and a₁ and a₂. This gives ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k}}} = T_{0}},\quad {{\sum\limits_{k = 0}^{2}{a_{k}S_{k + 1}}} = T_{1}},\quad {{\text{and}\quad {\sum\limits_{k = 0}^{2}{a_{k}S_{k + 2}}}} = T_{2}},$

where ${S_{0} = {{\sum\limits_{i = 1}^{7}x_{i}^{0}} = 7}};\quad {S_{1} = {{\sum\limits_{i = 1}^{7}x_{i}} = {\left( {{- 3} - 2 - 1 + 1 + 2 + 3} \right) = 0}}};$ ${S_{2} = {{\sum\limits_{i = 1}^{7}x_{i}^{2}} = {\left( {9 + 4 + 1 + 1 + 4 + 9} \right) = 28}}};$ ${S_{3} = {{\sum\limits_{i = 1}^{7}x_{i}^{3}} = {\left( {{- 27} - 8 - 1 + 0 + 1 + 8 + 27} \right) = 0}}};$ ${S_{4} = {{\sum\limits_{i = 1}^{7}x_{i}^{4}} = {\left( {81 + 16 + 1 + 0 + 1 + 16 + 81} \right) = 196}}};$ ${T_{0} = {{\sum\limits_{i = 1}^{7}y_{i}} = {\left( {4 + 2 + 3 + 0 - 1 - 2 - 5} \right) = 1}}};$ ${T_{1} = {{\sum\limits_{i = 1}^{7}{x_{i}y_{i}}} = {\left( {{- 12} - 4 - 3 + 0 - 1 - 4 - 15} \right) = {- 39}}}};$

and ${T_{2} = {{\sum\limits_{i = 1}^{7}{x_{i}^{2}y_{i}}} = {\left( {36 + 8 + 3 + 0 - 1 - 8 - 45} \right) = {- 7}}}},$

so that the normal equations become ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k}}} = {T_{0} = {1 = {{{7a_{0}} + {0a_{1}} + {28a_{2}}} = {{7a_{0}} + {28a_{2}}}}}}},{{\sum\limits_{k = 0}^{2}{a_{k}S_{k + 1}}} = {T_{1} = {{- 39} = {{0a_{0}} + {28a_{1}} + {0a_{2}}}}}},$

and ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k + 2}}} = {T_{2} = {{- 7} = {{{28a_{0}} + {0a_{1}} + {196a_{2}}} = {{28a_{0}} + {196a_{2}}}}}}},$

respectively, which imply (upon multiplying the first normal equation by 7 and then subtracting that from the third normal equation) that −14=−21a₀, that 28a₁=−39 (from the second normal equation), and (upon multiplying the first normal equation by 4 and then subtracting that from the third normal equation) that −11=84a₂, giving 3a₀=2, 28a₁=−39, and 84a₂−11, respectively. In other words, a₀=2/3, a₁=−39/28, and a₂=−11/84, so that, the second degree polynomial (the quadratic) that provides the best fit, in the least-squares sense, is ${{P_{2}(x)} = {{\frac{2}{3} - {\frac{39}{28}x} - {\frac{11}{84}x^{2}}} = {\frac{1}{84}\left( {56 - {117x} - {11x^{2}}} \right)}}},$

as shown in FIG. 22.

As shown in FIG. 23, for example, a second degree polynomial (a quadratic), ${{P_{2}(x)} = {{a_{0} + {a_{1}x} + {a_{2}x^{2}}} = {\sum\limits_{k = 0}^{2}{a_{k}x^{k}}}}},$

may be fit (in a least-squares sense) to the M=6 data points (0,4), (1,7), (2,10), (3,13), (4,16), and (5,19). The residuals ε_(i), for i=1, 2, . . . , 6, are schematically illustrated in FIG. 23. The three normal equations are ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k + j}}} = T_{j}},$

for j=0, 1, 2, where ${{\sum\limits_{i = 1}^{6}x_{i}^{k + j}} \equiv {S_{k + j}\quad \text{and}\quad {\sum\limits_{i = 1}^{6}{x_{i}^{j}y_{i}}}} \equiv T_{j}},$

respectively, for the three parameters a₀, and a₁ and a₂. This gives ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k}}} = T_{0}},\quad {{\sum\limits_{k = 0}^{2}{a_{k}S_{k + 1}}} = T_{1}},\quad {{\text{and}\quad {\sum\limits_{k = 0}^{2}{a_{k}S_{k + 2}}}} = T_{2}},$

where ${S_{0} = {{\sum\limits_{i = 1}^{6}x_{i}^{0}} = 6}};\quad {S_{1} = {{\sum\limits_{i = 1}^{6}x_{i}} = {\left( {0 + 1 + 2 + 3 + 4 + 5} \right) = 15}}};$ ${S_{2} = {{\sum\limits_{i = 1}^{6}x_{i}^{2}} = {\left( {1 + 4 + 9 + 16 + 25} \right) = 55}}};$ ${S_{3} = {{\sum\limits_{i = 1}^{6}x_{i}^{3}} = {\left( {0 + 1 + 8 + 27 + 64 + 125} \right) = 225}}};$ ${S_{4} = {{\sum\limits_{i = 1}^{6}x_{i}^{4}} = {\left( {0 + 1 + 16 + 81 + 256 + 625} \right) = 979}}};$ ${T_{0} = {{\sum\limits_{i = 1}^{6}y_{i}} = {\left( {4 + 7 + 10 + 13 + 16 + 19} \right) = 69}}};$ ${T_{1} = {{\sum\limits_{i = 1}^{6}{x_{i}y_{i}}} = {\left( {0 + 7 + 20 + 39 + 64 + 95} \right) = 225}}};$

and ${T_{2} = {{\sum\limits_{i = 1}^{6}{x_{i}^{2}y_{i}}} = {\left( {0 + 7 + 40 + 117 + 256 + 475} \right) = 895}}},$

so that the normal equations become ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k}}} = {T_{0} = {69 = {{6a_{0}} + {15a_{1}} + {55a_{2}}}}}},{{\sum\limits_{k = 0}^{2}{a_{k}S_{k + 1}}} = {T_{1} = {225 = {{15a_{0}} + {55a_{1}} + {225a_{2}}}}}},$

and ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k + 2}}} = {T_{2} = {895 = {{55a_{0}} + {225a_{1}} + {979a_{2}}}}}},$

respectively, which imply (upon multiplying the second normal equation by 4 and then subtracting that from the first normal equation multiplied by 10) that −210=−70a₁−350a₂, and (upon multiplying the second normal equation by 11 and then subtracting that from the third normal equation multiplied by 3) that 210=70a₁+66a₂. However, adding these last two results together shows that 0=a₂. Furthermore, 3=a₁. Therefore, using the fact that 3=a₁ and 0=a₂, the normal equations become ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k}}} = {T_{0} = {69 = {{6a_{0}} + 45}}}},{{\sum\limits_{k = 0}^{2}{a_{k}S_{k + 1}}} = {T_{1} = {225 = {{15a_{0}} + 165}}}},$

and ${{\sum\limits_{k = 0}^{2}{a_{k}S_{k + 2}}} = {T_{2} = {895 = {{55a_{0}} + 675}}}},$

respectively, which all imply that 4=a₀. In other words, a₀=4, a₁=3, and a₂=0, so that, the second degree polynomial (the quadratic) that provides the best fit, in the least-squares sense, is P₂(x)=4+3x+0x²=4+3x, which is really just a straight line, as shown in FIG. 23. The residuals ε_(i), for i=1, 2, . . . , 6, all vanish identically in this case, as schematically illustrated in FIG. 23.

In various other alternative illustrative embodiments, samples may be collected for M data points (x_(i),y_(i)), where i=1, 2, . . . , M, and a linearly independent set of N+1 functions ƒ_(j)(x), for j=0, 1, 2, . . . , N, ${{y(x)} = {{{a_{0}{f_{0}(x)}} + {a_{1}{f_{1}(x)}} + \ldots + {a_{j}{f_{j}(x)}} + \ldots + {a_{N}{f_{N}(x)}}} = {\sum\limits_{j = 0}^{N}{a_{j}{f_{j}(x)}}}}},$

may be fit (in a non-polynomial least-squares sense) to the M data points (x_(i),y_(i)). For example, 100 time data points (M=100) may be taken relating the pyrometer trace reading p, the lamp power trace reading f and/or the tube temperature trace reading T, during a rapid thermal processing step, to the effective yield t of workpieces emerging from the rapid thermal processing step, resulting in the M data points (p_(i),t_(i)), (f_(i),t_(i)), and/or (T_(i),t_(i)). The values may be the actually measured values of the rapid thermal processing tool variables and/or rapid thermal processing parameters, or ratios of actually measured values (normalized to respective reference setpoints), or logarithms of such ratios, for example. In one illustrative embodiment, the number N+1 of the linearly independent set of basis functions f_(j)(x) is at least 10 times smaller than M. Representative values for the number M of data points lie in a range of about 10 to about 1000, and representative values for the degree N of the polynomials is in a range of about 1 to about 10.

The function F(a₀, a₁, . . . ,a_(N)) may be minimized as follows. The function F(a₀,a₁, . . . ,a_(N)) is given by ${F\left( {a_{0},a_{1},\ldots \quad,a_{N}} \right)} = {{\sum\limits_{i = 1}^{M}ɛ_{i}^{2}} = {\sum\limits_{i = 1}^{M}\left\lbrack {{y\left( x_{i} \right)} - y_{i}} \right\rbrack^{2}}}$

and setting the partial derivatives of F(a₀,a₁, . . . ,a_(N)) with respect to a_(j), for j=0, 1, . . . , N, equal to zero gives ${\frac{\partial{F\left( {a_{0},a_{1},\ldots \quad,a_{N}} \right)}}{\partial a_{j}} = {{2{\sum\limits_{i = 1}^{M}{\left\lbrack {{y\left( x_{i} \right)} - y_{i}} \right\rbrack x_{i}^{j}}}} = {{2{\sum\limits_{i = 1}^{M}{\left\lbrack {{\sum\limits_{k = 0}^{N}{a_{k}{f_{k}\left( x_{i} \right)}}} - y_{i}} \right\rbrack {f_{j}\left( x_{i} \right)}}}} = 0}}},$

for j=0, 1, . . . , N, since ƒ_(j)(x_(i)) is the coefficient of a_(j) in the representation ${y\left( x_{i} \right)} = {\sum\limits_{k = 0}^{N}{a_{k}{{f_{k}\left( x_{i} \right)}.}}}$

Simplifying gives ${{\sum\limits_{i = 1}^{M}{\left\lbrack {\sum\limits_{k = 0}^{N}{a_{k}{f_{k}\left( x_{i} \right)}}} \right\rbrack {f_{j}\left( x_{i} \right)}}} = {{{\sum\limits_{k = 0}^{N}{a_{k}\left\lbrack {\sum\limits_{i = 1}^{M}{{f_{k}\left( x_{i} \right)}{f_{j}\left( x_{i} \right)}}} \right\rbrack}} \equiv {\sum\limits_{k = 0}^{N}{a_{k}S_{k,j}}}} = {{\sum\limits_{i = 1}^{M}{{f_{j}\left( x_{i} \right)}y_{i}}} \equiv T_{j}}}},$

for j=0, 1, . . . , N, where ${{\sum\limits_{i = 1}^{M}{{f_{k}\left( x_{i} \right)}{f_{j}\left( x_{i} \right)}}} \equiv {S_{k,j}\quad \text{and}\quad {\sum\limits_{i = 1}^{M}{{f_{j}\left( x_{i} \right)}y_{i}}}} \equiv T_{j}},$

respectively. There are N+1 equations ${{\sum\limits_{k = 0}^{N}{a_{k}S_{k,j}}} = T_{j}},$

for j=0, 1, . . . , N, also known as the normal equations, for the N+1 unknown parameters a_(k), for k=0, 1, . . . , N, readily yielding a solution, provided that the determinant of the normal equations is not zero. This may be demonstrated by showing that the homogeneous equations ${\sum\limits_{k = 0}^{N}{a_{k}S_{k,j}}} = 0$

only have the trivial solution a_(k)=0, for k=0, 1, . . . N, which may be shown as follows. Multiply the jth homogeneous equation by a_(j) and sum over all j, ${{\sum\limits_{j = 0}^{N}{a_{j}{\sum\limits_{k = 0}^{N}{a_{k}S_{k,j}}}}} = {{\sum\limits_{j = 0}^{N}{a_{j}{\sum\limits_{k = 0}^{N}{a_{k}{\sum\limits_{i = 1}^{M}{{f_{k}\left( x_{i} \right)}{f_{j}\left( x_{i} \right)}}}}}}} = {\sum\limits_{i = 1}^{M}{\left( {\sum\limits_{k = 0}^{N}{a_{k}{f_{k}\left( x_{i} \right)}}} \right)\left( {\sum\limits_{j = 0}^{N}{a_{j}{f_{j}\left( x_{i} \right)}}} \right)}}}},$

but ${{\sum\limits_{i = 1}^{M}{\left( {\sum\limits_{k = 0}^{N}{a_{k}{f_{k}\left( x_{i} \right)}}} \right)\left( {\sum\limits_{j = 0}^{N}{a_{j}{f_{j}\left( x_{i} \right)}}} \right)}} = {{\sum\limits_{i = 1}^{M}\left( {y\left( x_{i} \right)} \right)^{2}} = 0}},$

which would imply that y(x_(i))≡0, and, hence, that a_(k)=0, for k=0, 1, . . . , N, the trivial solution. Therefore, the determinant of the normal equations is not zero, and the normal equations may be solved for the N+1 parameters a_(k), for k=0, 1, . . . , N, the coefficients of the non-polynomial least-squares representation ${y(x)} = {\sum\limits_{j = 0}^{N}{a_{j}{f_{j}(x)}}}$

that may be fit to the M data points (x_(i),y_(i)), using the linearly independent set of N+1 functions ƒ_(j)(x) as the basis for the non-polynomial least-squares representation ${y(x)} = {\sum\limits_{j = 0}^{N}{a_{j}{{f_{j}(x)}.}}}$

If the data points (x_(i),y_(i)) are not equally reliable for all M, it may be desirable to weight the data by using non-negative weighting factors w_(i). The function F(a₀,a₁, . . . ,a_(N)) may be minimized as follows. The function F(a₀,a₁, . . . , a_(N)) is given by ${F\left( {a_{0},a_{1},\ldots \quad,a_{N}} \right)} = {{\sum\limits_{i = 1}^{M}{w_{i}ɛ_{i}^{2}}} = {\sum\limits_{i = 1}^{M}{w_{i}\left\lbrack {{y\left( x_{i} \right)} - y_{i}} \right\rbrack}^{2}}}$

and setting the partial derivatives of F(a₀,a₁, . . . ,a_(N)) with respect to a_(j), for j=0, 1, . . . , N, equal to zero gives ${\frac{\partial{F\left( {a_{0},a_{1},\ldots \quad,a_{N}} \right)}}{\partial a_{j}} = {{2{\sum\limits_{i = 1}^{M}{{w_{i}\left\lbrack {{y\left( x_{i} \right)} - y_{i}} \right\rbrack}x_{i}^{j}}}} = {{2{\sum\limits_{i = 1}^{M}{{w_{i}\left\lbrack {{\sum\limits_{k = 0}^{N}{a_{k}{f_{k}\left( x_{i} \right)}}} - y_{i}} \right\rbrack}{f_{j}\left( x_{i} \right)}}}} = 0}}},$

for j=0, 1, . . . , N, since ƒ_(j)(x_(i)) is the coefficient of a_(j) in the representation ${y\left( x_{i} \right)} = {\sum\limits_{k = 0}^{N}{a_{k}{{f_{k}\left( x_{i} \right)}.}}}$

Simplifying gives ${{\sum\limits_{i = 1}^{M}{{w_{i}\left\lbrack {\sum\limits_{k = 0}^{N}{a_{k}{f_{k}\left( x_{i} \right)}}} \right\rbrack}{f_{j}\left( x_{i} \right)}}} = {{\sum\limits_{k = 0}^{N}{a_{k}\left\lbrack {\sum\limits_{i = 1}^{M}{w_{i}{f_{k}\left( x_{i} \right)}{f_{j}\left( x_{i} \right)}}} \right\rbrack}} = {\sum\limits_{i = 1}^{M}{w_{i}{f_{j}\left( x_{i} \right)}y_{i}}}}},$

or ${{\sum\limits_{k = 0}^{N}{a_{k}\left\lbrack {\sum\limits_{i = 1}^{M}{w_{i}{f_{k}\left( x_{i} \right)}{f_{j}\left( x_{i} \right)}}} \right\rbrack}} \equiv {\sum\limits_{k = 0}^{N}{a_{k}S_{k,j}}}} = {{\sum\limits_{i = 1}^{M}{w_{i}{f_{j}\left( x_{i} \right)}y_{i}}} \equiv T_{j}}$

for j=0, 1, . . . , N, where ${{\sum\limits_{i = 1}^{M}{w_{i}{f_{k}\left( x_{i} \right)}{f_{j}\left( x_{i} \right)}}} \equiv {S_{k,j}\quad \text{and}\quad {\sum\limits_{i = 1}^{M}{w_{i}{f_{j}\left( x_{i} \right)}y_{i}}}} \equiv T_{j}},$

respectively. There are N+1 equations ${{\sum\limits_{k = 0}^{N}{a_{k}S_{k,j}}} = T_{j}},$

for j=0, 1, . . . , N, also known as the normal equations, including the non-negative weighting factors w_(i), for the N+1 unknown parameters a_(k), for k=0, 1, . . . , N, readily yielding a solution, provided that the determinant of the normal equations is not zero. This may be demonstrated by showing that the homogeneous equations ${\sum\limits_{k = 0}^{N}{a_{k}S_{k,j}}} = 0$

only have the trivial solution a_(k)=0, for k=0, 1, . . . , N, which may be shown as follows. Multiply the jth homogeneous equation by a_(j) and sum over all j, $\begin{matrix} {{\sum\limits_{j = 0}^{N}{a_{j}{\sum\limits_{k = 0}^{N}{a_{k}S_{k,j}}}}} = {\sum\limits_{j = 0}^{N}{a_{j}{\sum\limits_{k = 0}^{N}{a_{k}{\sum\limits_{i = 1}^{M}{w_{i}{f_{k}\left( x_{i} \right)}{f_{j}\left( x_{i} \right)}}}}}}}} \\ {{= {\sum\limits_{i = 0}^{M}{w_{i}\left( {\sum\limits_{k = 0}^{N}{a_{k}{f_{k}\left( x_{i} \right)}}} \right)\left( {\sum\limits_{j = 0}^{N}{a_{j}{f_{j}\left( x_{i} \right)}}} \right)}}},} \end{matrix}$

but ${{\sum\limits_{i = 1}^{M}{{w_{i}\left( {\sum\limits_{k = 0}^{N}{a_{k}{f_{k}\left( x_{i} \right)}}} \right)}\left( {\sum\limits_{j = 0}^{N}{a_{j}{f_{j}\left( x_{i} \right)}}} \right)}} = {{\sum\limits_{i = 1}^{M}{w_{i}\left( {y\left( x_{i} \right)} \right)}^{2}} = 0}},$

which would imply that y(x_(i))≡0, and, hence, that a_(k)=0, for k=0, 1, . . . , N, the trivial solution. Therefore, the determinant of the normal equations is not zero, and the normal equations, including the non-negative weighting factors w_(i), may be solved for the N+1 parameters a_(k), for k=0, 1, . . . , N, the coefficients of the non-polynomial least-squares representation ${y(x)} = {\sum\limits_{j = 0}^{N}{a_{j}{f_{j}(x)}}}$

that may be fit to the M data points (x_(i),y_(i)), using the linearly independent set of N+1 functions ƒ_(j)(x) as the basis for the non-polynomial least-squares representation ${{y(x)} = {\sum\limits_{j = 0}^{N}{a_{j}{f_{j}(x)}}}},$

and including the non-negative weighting factors w_(i).

Any of the above-disclosed embodiments of a method according to the present invention enables the use of parametric measurements sent from measuring tools to make supervisory processing adjustments, either manually and/or automatically, to improve and/or better control the yield. Additionally, any of the above-disclosed embodiments of a method of manufacturing according to the present invention enables semiconductor device fabrication with increased device accuracy and precision, increased efficiency and increased device yield, enabling a streamlined and simplified process flow, thereby decreasing the complexity and lowering the costs of the manufacturing process and increasing throughput.

The particular embodiments disclosed above are illustrative only, as the invention may be modified and practiced in different but equivalent manners apparent to those skilled in the art having the benefit of the teachings herein. Furthermore, no limitations are intended to the details of construction or design herein shown, other than as described in the claims below. It is therefore evident that the particular embodiments disclosed above may be altered or modified and all such variations are considered within the scope and spirit of the invention. In particular, every range of values (of the form, “from about a to about b,” or, equivalently, “from approximately a to b,” or, equivalently, “from approximately a-b”) disclosed herein is to be understood as referring to the power set (the set of all subsets) of the respective range of values, in the sense of Georg Cantor. Accordingly, the protection sought herein is as set forth in the claims below. 

What is claimed:
 1. A method comprising: measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step; modeling the at least one characteristic parameter measured using a first-principles radiation model incorporating at least one proportional-integral-derivative (PID) controller having at least one tuning parameter; and applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.
 2. The method of claim 1, wherein measuring the at least one parameter characteristic of the processing performed on the workpiece in the processing step comprises monitoring the at least one characteristic parameter using an advanced process control (APC) system.
 3. The method of claim 2, wherein monitoring the at least one characteristic parameter using the advanced process control (APC) system comprises using the advanced process control (APC) system to monitor at least one tool variable of a rapid thermal processing tool during the rapid thermal processing step.
 4. The method of claim 1, wherein using the first-principles radiation model incorporating the at least one proportional-integral-derivative (PID) controller having the at least one tuning parameter comprises using the first-principles radiation model incorporating at least one closed-loop proportional-integral-derivative (PID) controller having the at least one tuning parameter.
 5. The method of claim 1, wherein applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step comprises tuning the at least one tuning parameter to improve the rapid thermal processing performed in the rapid thermal processing step.
 6. The method of claim 1, further comprising processing at least one subsequent wafer in accordance with the modified rapid thermal processing.
 7. A computer-readable, program storage device, encoded with instructions that, when executed by a computer, perform a method comprising: measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step; modeling the at least one characteristic parameter measured using a first-principles radiation model incorporating at least one proportional-integral-derivative (PID) controller having at least one tuning parameter; and applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.
 8. The device of claim 7, wherein measuring the at least one parameter characteristic of the processing performed on the workpiece in the processing step comprises monitoring the at least one characteristic parameter using an advanced process control (APC) system.
 9. The device of claim 8, wherein monitoring the at least one characteristic parameter using the advanced process control (APC) system comprises using the advanced process control (APC) system to monitor at least one tool variable of a rapid thermal processing tool during the rapid thermal processing step.
 10. The device of claim 7, wherein using the first-principles radiation model incorporating the at least one proportional-integral-derivative (PID) controller having the at least one tuning parameter comprises using the first-principles radiation model incorporating at least one closed-loop proportional-integral-derivative (PID) controller having the at least one tuning parameter.
 11. The device of claim 7, wherein applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step comprises tuning the at least one tuning parameter to improve the rapid thermal processing performed in the rapid thermal processing step.
 12. The device of claim 7, wherein the method further comprises directing a tool to process at least one subsequent wafer in accordance with the modified rapid thermal processing.
 13. A computer programmed to perform a method comprising: measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step; modeling the at least one characteristic parameter measured using a first-principles radiation model incorporating at least one proportional-integral-derivative (PID) controller having at least one tuning parameter; and applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.
 14. The computer of claim 13, wherein measuring the at least one parameter characteristic of the processing performed on the workpiece in the processing step comprises monitoring the at least one characteristic parameter using an advanced process control (APC) system.
 15. The computer of claim 14, wherein monitoring the at least one characteristic parameter using the advanced process control (APC) system comprises using the advanced process control (APC) system to monitor at least one tool variable of a rapid thermal processing tool during the rapid thermal processing step.
 16. The computer of claim 13, wherein using the first-principles radiation model incorporating the at least one proportional-integral-derivative (PID) controller having the at least one tuning parameter comprises using the first-principles radiation model incorporating at least one closed-loop proportional-integral-derivative (PID) controller having the at least one tuning parameter.
 17. The computer of claim 13, wherein applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step comprises tuning the at least one tuning parameter to improve the rapid thermal processing performed in the rapid thermal processing step.
 18. The device of claim 13, wherein the method further comprises directing a tool to process at least one subsequent wafer in accordance with the modified rapid thermal processing.
 19. A method comprising: measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step; modeling the at least one characteristic parameter measured using a first-principles radiation model incorporating at least one proportional-integral-derivative (PID) controller having a plurality of tuning parameters; and applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.
 20. The method of claim 19, wherein measuring the at least one parameter characteristic of the processing performed on the workpiece in the processing step comprises monitoring the at least one characteristic parameter using an advanced process control (APC) system.
 21. The method of claim 20, wherein monitoring the at least one characteristic parameter using the advanced process control (APC) system comprises using the advanced process control (APC) system to monitor at least one tool variable of a rapid thermal processing tool during the rapid thermal processing step.
 22. The method of claim 19, wherein modeling the at least one characteristic parameter measured using the first-principles radiation model incorporating at least one proportional-integral-derivative (PID) controller having the plurality of tuning parameters comprises using the first-principles radiation model incorporating a plurality of proportional-integral-derivative (PID) controllers each having the plurality of tuning parameters.
 23. The method of claim 22, wherein using the first-principles radiation model incorporating the plurality of proportional-integral-derivative (PID) controllers each having the plurality of tuning parameters comprises using the first-principles radiation model incorporating a plurality of closed-loop proportional-integral-derivative (PID) controllers each having the plurality of tuning parameters.
 24. The method of claim 22, wherein applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step comprises tuning the plurality of tuning parameters to improve the rapid thermal processing performed in the rapid thermal processing step.
 25. The method of claim 20, wherein modeling the at least one characteristic parameter measured using the first-principles radiation model incorporating at least one proportional-integral-derivative (PID) controller having the plurality of tuning parameters comprises using the first-principles radiation model incorporating a plurality of proportional-integral-derivative (PID) controllers each having the plurality of tuning parameters.
 26. The method of claim 25, wherein applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step comprises tuning the plurality of tuning parameters to improve the rapid thermal processing performed in the rapid thermal processing step.
 27. The method of claim 21, wherein modeling the at least one characteristic parameter measured using the first-principles radiation model incorporating at least one proportional-integral-derivative (PID) controller having the plurality of tuning parameters comprises using the first-principles radiation model incorporating a plurality of proportional-integral-derivative (PID) controllers each having the plurality of tuning parameters.
 28. The method of claim 27, wherein applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step comprises tuning the plurality of tuning parameters to improve the rapid thermal processing performed in the rapid thermal processing step.
 29. The method of claim 22, further comprising processing at least one subsequent wafer in accordance with the modified rapid thermal processing.
 30. A system comprising: a tool for measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step; a computer for modeling the at least one characteristic parameter measured using a first-principles radiation model and a proportional-integral-derivative (PID) controller having at least one tuning parameter for applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.
 31. The system of claim 30, wherein the tool for measuring the at least one parameter characteristic of the processing performed on the workpiece in the processing step comprises a monitor for monitoring the at least one characteristic parameter using an advanced process control (APC) system.
 32. The system of claim 31, wherein the advanced process control (APC) system monitors at least one tool variable of a rapid thermal processing tool during the rapid thermal processing step.
 33. The system of claim 30, wherein the controller further comprises at least one closed-loop proportional-integral-derivative (PID) controller having the at least one tuning parameter.
 34. The system of claim 30, wherein the controller applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step tunes the at least one tuning parameter to improve the rapid thermal processing performed in the rapid thermal processing step.
 35. The system of claim 30, further comprising a tool for processing at least one subsequent wafer in accordance with the modified rapid thermal processing.
 36. A device comprising: means for measuring at least one parameter characteristic of rapid thermal processing performed on a workpiece in a rapid thermal processing step; means for modeling the at least one characteristic parameter measured using a first-principles radiation model incorporating at least one proportional-integral-derivative (PID) controller having at least one tuning parameter; and means for applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step.
 37. The device of claim 36, wherein the means for measuring the at least one parameter characteristic of the processing performed on the workpiece in the processing step comprises means for monitoring the at least one characteristic parameter using an advanced process control (APC) system.
 38. The device of claim 37, wherein the means for monitoring the at least one characteristic parameter using the advanced process control (APC) system comprises using the advanced process control (APC) system to monitor at least one tool variable of a rapid thermal processing tool during the rapid thermal processing step.
 39. The device of claim 36, wherein the means for using the first-principles radiation model incorporating the at least one proportional-integral-derivative (PID) controller having the at least one tuning parameter comprises the means for using the first-principles radiation model incorporating at least one closed-loop proportional-integral-derivative (PID) controller having the at least one tuning parameter.
 40. The device of claim 36, wherein the means for applying the first-principles radiation model to modify the rapid thermal processing performed in the rapid thermal processing step comprises means for tuning the at least one tuning parameter to improve the rapid thermal processing performed in the rapid thermal processing step.
 41. The device of claim 36, further comprising means for processing at least one subsequent wafer in accordance with the modified rapid thermal processing. 